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) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_the... | Mathlib/FieldTheory/Minpoly/Field.lean | 270 | 272 | theorem coeff_zero_ne_zero (hx : IsIntegral A x) (h : x ≠ 0) : coeff (minpoly A x) 0 ≠ 0 := by |
contrapose! h
simpa only [hx, coeff_zero_eq_zero] using 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, 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 | 1,587 | 1,591 | theorem Tendsto.prod_map_prod_atTop [SemilatticeSup γ] {F : Filter α} {G : Filter β} {f : α → γ}
{g : β → γ} (hf : Tendsto f F atTop) (hg : Tendsto g G atTop) :
Tendsto (Prod.map f g) (F ×ˢ G) atTop := by |
rw [← prod_atTop_atTop_eq]
exact hf.prod_map hg
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 569 | 571 | theorem symmDiff_eq_iff_sdiff_eq (ha : a ≤ c) : a ∆ b = c ↔ c \ a = b := by |
rw [← symmDiff_of_le ha]
exact ((symmDiff_right_involutive a).toPerm _).apply_eq_iff_eq_symm_apply.trans eq_comm
|
/-
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 | 1,369 | 1,373 | theorem tail_ofSeq (s : Seq α) : tail (ofSeq s) = ofSeq s.tail := by |
simp only [tail, destruct_ofSeq, map_pure', flatten_pure]
induction' s using Seq.recOn with x s <;> simp only [ofSeq, Seq.tail_nil, Seq.head_nil,
Option.map_none', Seq.tail_cons, Seq.head_cons, Option.map_some']
· rfl
|
/-
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 | 394 | 402 | theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) {f : α → ℝ≥0∞}
(hfm : AEMeasurable f μ) : Tendsto (∫⁻ x in φ ·, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) := by |
have lim₁ := lintegral_tendsto_of_monotone_of_nat hφ.biInter_Ici_aecover
(fun i j hij => biInter_subset_biInter_left (Ici_subset_Ici.mpr hij)) hfm
have lim₂ := lintegral_tendsto_of_monotone_of_nat hφ.biUnion_Iic_aecover
(fun i j hij => biUnion_subset_biUnion_left (Iic_subset_Iic.mpr hij)) hfm
refine tend... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 344 | 344 | theorem succ_aleph0 : succ ℵ₀ = aleph 1 := by | rw [← aleph_zero, ← aleph_succ, Ordinal.succ_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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 704 | 704 | theorem frontier_univ : frontier (univ : Set X) = ∅ := by | simp [frontier]
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
... | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | 137 | 212 | theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
(hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 := by |
-- First define some abbreviations.
letI := B.finite
let P := minpoly R B.gen
obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
have finrank_K_L : FiniteDimensional.finrank K L = B.dim := B.finrank
have deg_K_P : (minpoly K B.gen).natDegree = B.dim := B.natDegree_minpoly
have deg_R_P : P.natD... |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.GroupTheory.PresentedGroup
import Mathlib.GroupTheory.Coxeter.Matrix
/-!
# Coxeter groups and Coxeter syst... | Mathlib/GroupTheory/Coxeter/Basic.lean | 208 | 209 | theorem simple_mul_simple_cancel_left {w : W} (i : B) : s i * (s i * w) = w := by |
simp [← mul_assoc]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 25 | 30 | theorem support_mul [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G) :
(a * b).support ⊆ a.support * b.support := by |
rw [MonoidAlgebra.mul_def]
exact support_sum.trans <| biUnion_subset.2 fun _x hx ↦
support_sum.trans <| biUnion_subset.2 fun _y hy ↦
support_single_subset.trans <| singleton_subset_iff.2 <| mem_image₂_of_mem hx hy
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce... | Mathlib/Algebra/MvPolynomial/Expand.lean | 59 | 61 | theorem expand_one : expand 1 = AlgHom.id R (MvPolynomial σ R) := by |
ext1 f
rw [expand_one_apply, AlgHom.id_apply]
|
/-
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.Algebra.Group.Support
import Mathlib.Order.WellFoundedSet
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e126... | Mathlib/RingTheory/HahnSeries/Basic.lean | 382 | 386 | theorem support_embDomain_subset {f : Γ ↪o Γ'} {x : HahnSeries Γ R} :
support (embDomain f x) ⊆ f '' x.support := by |
intro g hg
contrapose! hg
rw [mem_support, embDomain_notin_image_support hg, Classical.not_not]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 611 | 613 | theorem comap_id : comap (RingHom.id R) = ContinuousMap.id _ := by |
ext
rfl
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import ... | Mathlib/Data/List/Chain.lean | 335 | 336 | theorem chain'_pair {x y} : Chain' R [x, y] ↔ R x y := by |
simp only [chain'_singleton, chain'_cons, and_true_iff]
|
/-
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.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "l... | Mathlib/Order/SuccPred/Basic.lean | 315 | 320 | theorem isMax_iterate_succ_of_eq_of_ne {n m : ℕ} (h_eq : succ^[n] a = succ^[m] a)
(h_ne : n ≠ m) : IsMax (succ^[n] a) := by |
rcases le_total n m with h | h
· exact isMax_iterate_succ_of_eq_of_lt h_eq (lt_of_le_of_ne h h_ne)
· rw [h_eq]
exact isMax_iterate_succ_of_eq_of_lt h_eq.symm (lt_of_le_of_ne h h_ne.symm)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 182 | 183 | theorem coe_iInf {ι : Sort*} (S : ι → IntermediateField F E) : (↑(iInf S) : Set E) = ⋂ i, S i := by |
simp [iInf]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 104 | 104 | theorem prod_univ {s : Set α} : s ×ˢ (univ : Set β) = Prod.fst ⁻¹' s := by | simp [prod_eq]
|
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd0... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 285 | 286 | theorem mem_row_iff {μ : YoungDiagram} {i : ℕ} {c : ℕ × ℕ} : c ∈ μ.row i ↔ c ∈ μ ∧ c.fst = i := by |
simp [row]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_e... | Mathlib/FieldTheory/PrimitiveElement.lean | 282 | 292 | theorem FiniteDimensional.of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by |
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
have := isAlgebraic_of_finite_intermediateField F E
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(Algebra.IsIntegral.isIntegral _)
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have... |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.WellFound... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 682 | 684 | theorem pow_eq_closure_pow_set (s : AddSubmonoid R) (n : ℕ) :
s ^ n = closure ((s : Set R) ^ n) := by |
rw [← closure_pow, closure_eq]
|
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
imp... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 45 | 56 | theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
(A / C).card * B.card ≤ (A / B).card * (B / C).card := by |
rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
refine card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ ?_)
fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
((mem_bipartiteBelow _).1 hu).2.symm.trans ?_
obtain ⟨a, ha, c, hc, rfl⟩ := mem_div.1 hx
refine card_le_card_of_inj_on (fun... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 529 | 532 | theorem support_le_prod_of_mem {l : List (Perm α)} (h : f ∈ l) (hl : l.Pairwise Disjoint) :
f.support ≤ l.prod.support := by |
intro x hx
rwa [mem_support, ← eq_on_support_mem_disjoint h hl _ hx, ← mem_support]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 1,003 | 1,004 | theorem nsmul_singleton (a : α) (n) : n • ({a} : Multiset α) = replicate n a := by |
rw [← replicate_one, nsmul_replicate, mul_one]
|
/-
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.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Spe... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 372 | 375 | theorem InnerProductSpace.volume_closedBall (x : E) (r : ℝ) :
volume (Metric.closedBall x r) = (.ofReal r) ^ finrank ℝ E *
.ofReal (sqrt π ^ finrank ℝ E / Gamma (finrank ℝ E / 2 + 1)) := by |
rw [addHaar_closedBall_eq_addHaar_ball, InnerProductSpace.volume_ball _]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 137 | 139 | theorem postComp_φ : (f.postComp h).φ = f.φ ≫ h := by |
unfold φ postComp
simp only [add_comp, sum_comp, assoc]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 397 | 400 | theorem multichoose_two (k : ℕ) : multichoose 2 k = k + 1 := by |
induction' k with k IH; · simp
rw [multichoose, IH]
simp [Nat.add_comm, succ_eq_add_one]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mat... | Mathlib/SetTheory/Ordinal/Exponential.lean | 78 | 79 | theorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a := by |
rw [← succ_zero, opow_succ]; simp only [opow_zero, one_mul]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 145 | 145 | theorem sMod_mod (p i : ℕ) : sMod p i % (2 ^ p - 1) = sMod p i := by | cases i <;> simp [sMod]
|
/-
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.Algebra.Defs
import Mathlib.Algebra.Ring.Subring.Basic
#align_import field_theory.subfield from "leanprover-community/mathlib"@"28aa996fc6fb4317f008... | Mathlib/Algebra/Field/Subfield.lean | 332 | 335 | theorem zpow_mem {x : K} (hx : x ∈ s) (n : ℤ) : x ^ n ∈ s := by |
cases n
· simpa using s.pow_mem hx _
· simpa [pow_succ'] using s.inv_mem (s.mul_mem hx (s.pow_mem hx _))
|
/-
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 | 498 | 500 | theorem HasFDerivAt.unique (h₀ : HasFDerivAt f f₀' x) (h₁ : HasFDerivAt f f₁' x) : f₀' = f₁' := by |
rw [← hasFDerivWithinAt_univ] at h₀ h₁
exact uniqueDiffWithinAt_univ.eq h₀ h₁
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Cho... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 128 | 205 | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
{f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
(hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
‖iteratedFDerivWithin 𝕜 n (fun y => B... |
/- We reduce the bound to the case where all spaces live in the same universe (in which we
already have proved the result), by using linear isometries between the spaces and their `ULift`
to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
let Du : Type max uD uE ... |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | Mathlib/Data/List/Sym.lean | 46 | 47 | theorem sym2_eq_nil_iff {xs : List α} : xs.sym2 = [] ↔ xs = [] := by |
cases xs <;> simp [List.sym2]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 35 | 41 | theorem Ioi_add_bij : BijOn (· + d) (Ioi a) (Ioi (a + d)) := by |
refine
⟨fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h =>
?_⟩
obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ioi.mp h).le
rw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h
exact ⟨a + c, h, by rw [add_right_comm]⟩
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.SplitIfs
#align_import data.nat.cast.defs from "leanprover-community/mathlib"@"a148d797a1094ab... | Mathlib/Data/Nat/Cast/Defs.lean | 159 | 162 | theorem cast_add [AddMonoidWithOne R] (m n : ℕ) : ((m + n : ℕ) : R) = m + n := by |
induction n with
| zero => simp
| succ n ih => rw [add_succ, cast_succ, ih, cast_succ, add_assoc]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 86 | 89 | theorem single_mem_gradeBy {R} [CommSemiring R] (f : M → ι) (m : M) (r : R) :
Finsupp.single m r ∈ gradeBy R f (f m) := by |
intro x hx
rw [Finset.mem_singleton.mp (Finsupp.support_single_subset hx)]
|
/-
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 | 661 | 664 | theorem finSepDegree_adjoin_simple_le_finrank (α : E) (halg : IsAlgebraic F α) :
finSepDegree F F⟮α⟯ ≤ finrank F F⟮α⟯ := by |
haveI := adjoin.finiteDimensional halg.isIntegral
exact Nat.le_of_dvd finrank_pos <| finSepDegree_adjoin_simple_dvd_finrank F E α
|
/-
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 | 475 | 482 | theorem Perm.diff_right {l₁ l₂ : List α} (t : List α) (h : l₁ ~ l₂) : l₁.diff t ~ l₂.diff t := by |
induction t generalizing l₁ l₂ h with simp only [List.diff]
| nil => exact h
| cons x t ih =>
simp only [elem_eq_mem, decide_eq_true_eq, Perm.mem_iff h]
split
· exact ih (h.erase _)
· exact ih h
|
/-
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.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equ... | Mathlib/Logic/Equiv/PartialEquiv.lean | 943 | 946 | theorem prod_trans {η : Type*} {ε : Type*} (e : PartialEquiv α β) (f : PartialEquiv β γ)
(e' : PartialEquiv δ η) (f' : PartialEquiv η ε) :
(e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') := by |
ext ⟨x, y⟩ <;> simp [ext_iff]; tauto
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 243 | 245 | theorem encard_diff_singleton_add_one (h : a ∈ s) :
(s \ {a}).encard + 1 = s.encard := by |
rw [← encard_insert_of_not_mem (fun h ↦ h.2 rfl), insert_diff_singleton, insert_eq_of_mem h]
|
/-
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.Order.SuccPred.LinearLocallyFinite
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.optional_sampling from "leanprover-com... | Mathlib/Probability/Martingale/OptionalSampling.lean | 51 | 58 | theorem condexp_stopping_time_ae_eq_restrict_eq_const
[(Filter.atTop : Filter ι).IsCountablyGenerated] (h : Martingale f ℱ μ)
(hτ : IsStoppingTime ℱ τ) [SigmaFinite (μ.trim hτ.measurableSpace_le)] (hin : i ≤ n) :
μ[f n|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] f i := by |
refine Filter.EventuallyEq.trans ?_ (ae_restrict_of_ae (h.condexp_ae_eq hin))
refine condexp_ae_eq_restrict_of_measurableSpace_eq_on hτ.measurableSpace_le (ℱ.le i)
(hτ.measurableSet_eq' i) fun t => ?_
rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.... | Mathlib/MeasureTheory/Group/Measure.lean | 874 | 876 | theorem haar_singleton [TopologicalGroup G] [BorelSpace G] (g : G) : μ {g} = μ {(1 : G)} := by |
convert measure_preimage_mul μ g⁻¹ _
simp only [mul_one, preimage_mul_left_singleton, inv_inv]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,084 | 1,088 | theorem prod_subtype_map_embedding {p : α → Prop} {s : Finset { x // p x }} {f : { x // p x } → β}
{g : α → β} (h : ∀ x : { x // p x }, x ∈ s → g x = f x) :
(∏ x ∈ s.map (Function.Embedding.subtype _), g x) = ∏ x ∈ s, f x := by |
rw [Finset.prod_map]
exact Finset.prod_congr rfl h
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 90 | 92 | theorem multiplicity_eq_factorization {n p : ℕ} (pp : p.Prime) (hn : n ≠ 0) :
multiplicity p n = n.factorization p := by |
simp [factorization, pp, padicValNat_def' pp.ne_one hn.bot_lt]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | Mathlib/CategoryTheory/Abelian/Opposite.lean | 124 | 126 | theorem kernel.ι_unop :
(kernel.ι g.unop).op = eqToHom (Opposite.op_unop _) ≫ cokernel.π g ≫ (kernelUnopOp g).inv := by |
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,227 | 1,228 | theorem map_add (μ ν : Measure α) {f : α → β} (hf : Measurable f) :
(μ + ν).map f = μ.map f + ν.map f := by | simp [← mapₗ_apply_of_measurable hf]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 195 | 200 | theorem IsReal.coe_embedding_apply {φ : K →+* ℂ} (hφ : IsReal φ) (x : K) :
(hφ.embedding x : ℂ) = φ x := by |
apply Complex.ext
· rfl
· rw [ofReal_im, eq_comm, ← Complex.conj_eq_iff_im]
exact RingHom.congr_fun hφ x
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
/-!
# Power ... | Mathlib/RingTheory/PowerBasis.lean | 270 | 272 | theorem constr_pow_algebraMap (pb : PowerBasis A S) {y : S'} (hy : aeval y (minpoly A pb.gen) = 0)
(x : A) : pb.basis.constr A (fun i => y ^ (i : ℕ)) (algebraMap A S x) = algebraMap A S' x := by |
convert pb.constr_pow_aeval hy (C x) <;> rw [aeval_C]
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 174 | 175 | theorem toPoly_eq_zero_iff (P : Cubic R) : P.toPoly = 0 ↔ P = 0 := by |
rw [← zero, toPoly_injective]
|
/-
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.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,424 | 1,425 | theorem mul_singleton_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : s * {a} ∈ 𝓝 a := by |
simpa only [one_mul] using mul_singleton_mem_nhds a h
|
/-
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 | 1,656 | 1,662 | theorem setToFun_top_smul_measure (hT : DominatedFinMeasAdditive (∞ • μ) T C) (f : α → E) :
setToFun (∞ • μ) T hT f = 0 := by |
refine setToFun_measure_zero' hT fun s _ hμs => ?_
rw [lt_top_iff_ne_top] at hμs
simp only [true_and_iff, Measure.smul_apply, ENNReal.mul_eq_top, eq_self_iff_true,
top_ne_zero, Ne, not_false_iff, not_or, Classical.not_not, smul_eq_mul] at hμs
simp only [hμs.right, Measure.smul_apply, mul_zero, smul_eq_mul]... |
/-
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.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 248 | 253 | theorem card_support_trinomial {k m n : ℕ} (hkm : k < m) (hmn : m < n) {x y z : R} (hx : x ≠ 0)
(hy : y ≠ 0) (hz : z ≠ 0) : card (support (C x * X ^ k + C y * X ^ m + C z * X ^ n)) = 3 := by |
rw [support_trinomial hkm hmn hx hy hz,
card_insert_of_not_mem
(mt mem_insert.mp (not_or_of_not hkm.ne (mt mem_singleton.mp (hkm.trans hmn).ne))),
card_insert_of_not_mem (mt mem_singleton.mp hmn.ne), card_singleton]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 109 | 112 | theorem hasBasis_nhds_of_ne_zero {x : Γ₀} (h : x ≠ 0) :
HasBasis (𝓝 x) (fun _ : Unit => True) fun _ => {x} := by |
rw [nhds_of_ne_zero h]
exact hasBasis_pure _
|
/-
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.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Set... | Mathlib/GroupTheory/GroupAction/Basic.lean | 525 | 529 | theorem orbitRel.Quotient.mem_orbit {a : α} {x : orbitRel.Quotient G α} :
a ∈ x.orbit ↔ Quotient.mk'' a = x := by |
induction x using Quotient.inductionOn'
rw [Quotient.eq'']
rfl
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "lea... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 855 | 888 | theorem UniformIntegrable.spec' (hp : p ≠ 0) (hp' : p ≠ ∞) (hf : ∀ i, StronglyMeasurable (f i))
(hfu : UniformIntegrable f p μ) {ε : ℝ} (hε : 0 < ε) :
∃ C : ℝ≥0, ∀ i, snorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε := by |
obtain ⟨-, hfu, M, hM⟩ := hfu
obtain ⟨δ, hδpos, hδ⟩ := hfu hε
obtain ⟨C, hC⟩ : ∃ C : ℝ≥0, ∀ i, μ { x | C ≤ ‖f i x‖₊ } ≤ ENNReal.ofReal δ := by
by_contra hcon; push_neg at hcon
choose ℐ hℐ using hcon
lift δ to ℝ≥0 using hδpos.le
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ snorm (f (ℐ C)) p... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 332 | 335 | theorem vsub_left_mem_direction_iff_mem {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p2 : P) :
p -ᵥ p2 ∈ s.direction ↔ p2 ∈ s := by |
rw [mem_direction_iff_eq_vsub_left hp]
simp
|
/-
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.Analysis.Convex.StrictConvexBetween
import Mathlib.Geometry.Euclidean.Basic
#align_import geometry.euclidean.sphere.basic from "leanprover-community/mathl... | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 119 | 121 | theorem Sphere.ne_iff {s₁ s₂ : Sphere P} :
s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius := by |
rw [← not_and_or, ← Sphere.ext_iff]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 595 | 600 | theorem condexpL1_of_aestronglyMeasurable' (hfm : AEStronglyMeasurable' m f μ)
(hfi : Integrable f μ) : condexpL1 hm μ f =ᵐ[μ] f := by |
rw [condexpL1_eq hfi]
refine EventuallyEq.trans ?_ (Integrable.coeFn_toL1 hfi)
rw [condexpL1CLM_of_aestronglyMeasurable']
exact AEStronglyMeasurable'.congr hfm (Integrable.coeFn_toL1 hfi).symm
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 43 | 46 | theorem braiding_naturality_left {X Y : ModuleCat R} (f : X ⟶ Y) (Z : ModuleCat R) :
f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by |
simp_rw [← id_tensorHom]
apply braiding_naturality
|
/-
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.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#ali... | Mathlib/RingTheory/Int/Basic.lean | 99 | 102 | theorem Int.Prime.dvd_pow {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :
p ∣ n.natAbs := by |
rw [Int.natCast_dvd, Int.natAbs_pow] at h
exact hp.dvd_of_dvd_pow h
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 738 | 750 | theorem filter_image_mk_not_isDiag [DecidableEq α] (s : Finset α) :
(((s ×ˢ s).image Sym2.mk).filter fun a : Sym2 α => ¬a.IsDiag) =
s.offDiag.image Sym2.mk := by |
ext z
induction z using Sym2.inductionOn
simp only [mem_image, mem_offDiag, mem_filter, Prod.exists, mem_product]
constructor
· rintro ⟨⟨a, b, ⟨ha, hb⟩, h⟩, hab⟩
rw [← h, Sym2.mk_isDiag_iff] at hab
exact ⟨a, b, ⟨ha, hb, hab⟩, h⟩
· rintro ⟨a, b, ⟨ha, hb, hab⟩, h⟩
rw [Ne, ← Sym2.mk_isDiag_iff, h]... |
/-
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.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 120 | 164 | theorem intervalIntegrable_cpow {r : ℂ} (h : 0 ≤ r.re ∨ (0 : ℝ) ∉ [[a, b]]) :
IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) μ a b := by |
by_cases h2 : (0 : ℝ) ∉ [[a, b]]
· -- Easy case #1: 0 ∉ [a, b] -- use continuity.
refine (ContinuousAt.continuousOn fun x hx => ?_).intervalIntegrable
exact Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_mem_of_not_mem hx h2)
rw [eq_false h2, or_false_iff] at h
rcases lt_or_eq_of_le h with... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 3,052 | 3,053 | theorem disjoint_add_left {s t u : Multiset α} :
Disjoint (s + t) u ↔ Disjoint s u ∧ Disjoint t u := by | simp [Disjoint, or_imp, forall_and]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometr... | Mathlib/Geometry/RingedSpace/Stalks.lean | 188 | 192 | theorem congr_point {X Y : PresheafedSpace.{_, _, v} C}
(α : X ⟶ Y) (x x' : X) (h : x = x') :
stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h]) =
eqToHom (show Y.stalk (α.base x) = Y.stalk (α.base x') by rw [h]) ≫ stalkMap α x' := by |
rw [stalkMap.congr α α rfl x x' h]
|
/-
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, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 472 | 475 | theorem det_updateRow_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateRow A i (A i + A j)) = det A := by |
simp [det_updateRow_add,
det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)]
|
/-
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 | 375 | 386 | theorem Measurable.measurable_of_countable_ne [MeasurableSingletonClass α] (hf : Measurable f)
(h : Set.Countable { x | f x ≠ g x }) : Measurable g := by |
intro t ht
have : g ⁻¹' t = g ⁻¹' t ∩ { x | f x = g x }ᶜ ∪ g ⁻¹' t ∩ { x | f x = g x } := by
simp [← inter_union_distrib_left]
rw [this]
refine (h.mono inter_subset_right).measurableSet.union ?_
have : g ⁻¹' t ∩ { x : α | f x = g x } = f ⁻¹' t ∩ { x : α | f x = g x } := by
ext x
simp (config := {... |
/-
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 | 240 | 243 | theorem TendstoUniformlyOn.comp (h : TendstoUniformlyOn F f p s) (g : γ → α) :
TendstoUniformlyOn (fun n => F n ∘ g) (f ∘ g) p (g ⁻¹' s) := by |
rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter] at h ⊢
simpa [TendstoUniformlyOn, comap_principal] using TendstoUniformlyOnFilter.comp h g
|
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 202 | 217 | theorem continuousAt_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) :
ContinuousAt (fun p : ℝ × ℝ => p.1 ^ p.2) p := by |
rw [ne_iff_lt_or_gt] at hp
cases hp with
| inl hp =>
rw [continuousAt_congr (rpow_eq_nhds_of_neg hp)]
refine ContinuousAt.mul ?_ (continuous_cos.continuousAt.comp ?_)
· refine continuous_exp.continuousAt.comp (ContinuousAt.mul ?_ continuous_snd.continuousAt)
refine (continuousAt_log ?_).comp co... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 309 | 319 | theorem condexpL2_indicator_ae_eq_smul (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
(x : E') :
condexpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x) =ᵐ[μ] fun a =>
(condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs (1 : ℝ)) : α → ℝ) a • x := by |
rw [indicatorConstLp_eq_toSpanSingleton_compLp hs hμs x]
have h_comp :=
condexpL2_comp_continuousLinearMap ℝ 𝕜 hm (toSpanSingleton ℝ x)
(indicatorConstLp 2 hs hμs (1 : ℝ))
rw [← lpMeas_coe] at h_comp
refine h_comp.trans ?_
exact (toSpanSingleton ℝ x).coeFn_compLp _
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 204 | 206 | theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
(weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t := by |
ext1 x; simp_rw [weightedSMul_apply]; congr 2
|
/-
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 | 1,997 | 2,005 | theorem isBigOWith_iff_exists_eq_mul (hc : 0 ≤ c) :
IsBigOWith c l u v ↔ ∃ φ : α → 𝕜, (∀ᶠ x in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v := by |
constructor
· intro h
use fun x => u x / v x
refine ⟨Eventually.mono h.bound fun y hy => ?_, h.eventually_mul_div_cancel.symm⟩
simpa using div_le_of_nonneg_of_le_mul (norm_nonneg _) hc hy
· rintro ⟨φ, hφ, h⟩
exact isBigOWith_of_eq_mul φ hφ h
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 50 | 50 | theorem Disjoint.symm : Disjoint f g → Disjoint g f := by | simp only [Disjoint, or_comm, imp_self]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,245 | 2,251 | theorem mk_eq_two_iff : #α = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α) = univ := by |
simp only [← @Nat.cast_two Cardinal, mk_eq_nat_iff_finset, Finset.card_eq_two]
constructor
· rintro ⟨t, ht, x, y, hne, rfl⟩
exact ⟨x, y, hne, by simpa using ht⟩
· rintro ⟨x, y, hne, h⟩
exact ⟨{x, y}, by simpa using h, x, y, hne, rfl⟩
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 246 | 247 | theorem preimage_neg_Ioc : -Ioc a b = Ico (-b) (-a) := by |
simp [← Ioi_inter_Iic, ← Ici_inter_Iio, inter_comm]
|
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 489 | 493 | theorem totalDegree_finset_prod {ι : Type*} (s : Finset ι) (f : ι → MvPolynomial σ R) :
(s.prod f).totalDegree ≤ ∑ i ∈ s, (f i).totalDegree := by |
refine le_trans (totalDegree_multiset_prod _) ?_
rw [Multiset.map_map]
rfl
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 562 | 564 | theorem lf_of_le_of_lf {x y z : PGame} (h₁ : x ≤ y) (h₂ : y ⧏ z) : x ⧏ z := by |
rw [← PGame.not_le] at h₂ ⊢
exact fun h₃ => h₂ (h₃.trans h₁)
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Eq... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 242 | 246 | theorem whiskerLeft_π_actLeft :
(R.X ◁ coequalizer.π _ _) ≫ actLeft P Q =
(α_ _ _ _).inv ≫ (P.actLeft ▷ Q.X) ≫ coequalizer.π _ _ := by |
erw [map_π_preserves_coequalizer_inv_colimMap (tensorLeft _)]
simp only [Category.assoc]
|
/-
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 | 757 | 759 | theorem prod.map_map {A₁ A₂ A₃ B₁ B₂ B₃ : C} [HasBinaryProduct A₁ B₁] [HasBinaryProduct A₂ B₂]
[HasBinaryProduct A₃ B₃] (f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) (h : A₂ ⟶ A₃) (k : B₂ ⟶ B₃) :
prod.map f g ≫ prod.map h k = prod.map (f ≫ h) (g ≫ k) := by | ext <;> simp
|
/-
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 | 1,413 | 1,422 | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m ∈ Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by |
have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
have h' := Real.exp_bound h4 hn
rw [h3] at h'
have h'' := (abs_sub_le_iff.1 h').1
have t := sub_le_iff_le_add'.1 h''
simpa [mul_div_assoc] using t
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.big_operators from "leanprover-community/mathlib"... | Mathlib/Data/Set/Pointwise/BigOperators.lean | 184 | 186 | theorem image_fintype_prod_pi [Fintype ι] (S : ι → Set α) :
(fun f : ι → α => ∏ i, f i) '' univ.pi S = ∏ i, S i := by |
simpa only [Finset.coe_univ] using image_finset_prod_pi Finset.univ S
|
/-
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.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 171 | 175 | theorem length_simple_mul_ne (w : W) (i : B) : ℓ (s i * w) ≠ ℓ w := by |
convert cs.length_mul_simple_ne w⁻¹ i using 1
· convert cs.length_inv ?_ using 2
simp
· 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.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,266 | 1,271 | theorem contDiffWithinAt_extend_coord_change' [ChartedSpace H M] (hf : f ∈ maximalAtlas I M)
(hf' : f' ∈ maximalAtlas I M) {x : M} (hxf : x ∈ f.source) (hxf' : x ∈ f'.source) :
ContDiffWithinAt 𝕜 ⊤ (f.extend I ∘ (f'.extend I).symm) (range I) (f'.extend I x) := by |
refine contDiffWithinAt_extend_coord_change I hf hf' ?_
rw [← extend_image_source_inter]
exact mem_image_of_mem _ ⟨hxf', hxf⟩
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mat... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 419 | 433 | theorem coeff_mul_natTrailingDegree_add_natTrailingDegree : (p * q).coeff
(p.natTrailingDegree + q.natTrailingDegree) = p.trailingCoeff * q.trailingCoeff := by |
rw [coeff_mul]
refine
Finset.sum_eq_single (p.natTrailingDegree, q.natTrailingDegree) ?_ fun h =>
(h (mem_antidiagonal.mpr rfl)).elim
rintro ⟨i, j⟩ h₁ h₂
rw [mem_antidiagonal] at h₁
by_cases hi : i < p.natTrailingDegree
· rw [coeff_eq_zero_of_lt_natTrailingDegree hi, zero_mul]
by_cases hj : j <... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 332 | 333 | theorem span_attach_biUnion [DecidableEq M] {α : Type*} (s : Finset α) (f : s → Finset M) :
span R (s.attach.biUnion f : Set M) = ⨆ x, span R (f x) := by | simp [span_iUnion]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 807 | 807 | theorem div_le_one_of_neg (hb : b < 0) : a / b ≤ 1 ↔ b ≤ a := by | rw [div_le_iff_of_neg hb, one_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.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 257 | 284 | theorem AffineTargetMorphismProperty.isLocalOfOpenCoverImply (P : AffineTargetMorphismProperty)
(hP : P.toProperty.RespectsIso)
(H : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y),
(∃ (𝒰 : Scheme.OpenCover.{u} Y) (_ : ∀ i, IsAffine (𝒰.obj i)),
∀ i : 𝒰.J, P (pullback.snd : (𝒰.pullbackCover f).obj i ⟶ 𝒰.obj ... |
refine ⟨hP, ?_, ?_⟩
· introv h
haveI : IsAffine _ := (topIsAffineOpen Y).basicOpenIsAffine r
delta morphismRestrict
rw [affine_cancel_left_isIso hP]
refine @H _ _ f ⟨Scheme.openCoverOfIsIso (𝟙 Y), ?_, ?_⟩ _ (Y.ofRestrict _) _ _
· intro i; dsimp; infer_instance
· intro i; dsimp
rwa [←... |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from ... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 461 | 477 | theorem pullbackConeOfLeftLift_fst :
pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst = s.fst := by |
-- Porting note: `ext` did not pick up `NatTrans.ext`
refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext _ _ <| funext fun x => ?_
· change pullback.lift _ _ _ ≫ pullback.fst = _
simp
· induction x using Opposite.rec' with | h x => ?_
change ((_ ≫ _) ≫ _ ≫ _) ≫ _ = _
simp_rw [Category.assoc]
... |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformFilterBasis
import Mathlib.RingTheory.Valuation.ValuationSubring
#align_i... | Mathlib/Topology/Algebra/Valuation.lean | 159 | 168 | theorem cauchy_iff {F : Filter R} : Cauchy F ↔
F.NeBot ∧ ∀ γ : Γ₀ˣ, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), (v (y - x) : Γ₀) < γ := by |
rw [toUniformSpace_eq, AddGroupFilterBasis.cauchy_iff]
apply and_congr Iff.rfl
simp_rw [Valued.v.subgroups_basis.mem_addGroupFilterBasis_iff]
constructor
· intro h γ
exact h _ (Valued.v.subgroups_basis.mem_addGroupFilterBasis _)
· rintro h - ⟨γ, rfl⟩
exact h γ
|
/-
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
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
impor... | Mathlib/Dynamics/PeriodicPts.lean | 693 | 695 | theorem pow_add_period_smul (n : ℕ) (m : M) (a : α) :
m ^ (n + period m a) • a = m ^ n • a := by |
rw [← pow_mod_period_smul, Nat.add_mod_right, pow_mod_period_smul]
|
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_lim... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 484 | 494 | theorem uniformCauchySeqOn_ball_of_deriv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
(hf : ∀ n : ι, ∀ y : 𝕜, y ∈ Metric.ball x r → HasDerivAt (f n) (f' n y) y)
(hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) := by |
simp_rw [hasDerivAt_iff_hasFDerivAt] at hf
rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter] at hf'
have hf' :
UniformCauchySeqOn (fun n => fun z => (1 : 𝕜 →L[𝕜] 𝕜).smulRight (f' n z)) l
(Metric.ball x r) := by
rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter]
exact hf'.one_smulRight
... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 1,044 | 1,045 | theorem isCompact_iff_isCompact_univ : IsCompact s ↔ IsCompact (univ : Set s) := by |
rw [Subtype.isCompact_iff, image_univ, Subtype.range_coe]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92... | Mathlib/LinearAlgebra/LinearPMap.lean | 899 | 912 | theorem mem_range_iff {f : E →ₗ.[R] F} {y : F} : y ∈ Set.range f ↔ ∃ x : E, (x, y) ∈ f.graph := by |
constructor <;> intro h
· rw [Set.mem_range] at h
rcases h with ⟨⟨x, hx⟩, h⟩
use x
rw [← h]
exact f.mem_graph ⟨x, hx⟩
cases' h with x h
rw [mem_graph_iff] at h
cases' h with x h
rw [Set.mem_range]
use x
simp only at h
rw [h.2]
|
/-
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 | 445 | 447 | theorem inner_smul_rotation_pi_div_two_right (x : V) (r : ℝ) :
⟪x, r • o.rotation (π / 2 : ℝ) x⟫ = 0 := by |
rw [real_inner_comm, inner_smul_rotation_pi_div_two_left]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import ... | Mathlib/Data/List/Chain.lean | 170 | 174 | theorem chain_iff_nthLe {R} {a : α} {l : List α} : Chain R a l ↔
(∀ h : 0 < length l, R a (nthLe l 0 h)) ∧
∀ (i) (h : i < length l - 1),
R (nthLe l i (by omega)) (nthLe l (i + 1) (by omega)) := by |
rw [chain_iff_get]; simp [nthLe]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 792 | 794 | theorem limsup_const_bot {f : Filter β} : limsup (fun _ : β => (⊥ : α)) f = (⊥ : α) := by |
rw [limsup_eq, eq_bot_iff]
exact sInf_le (eventually_of_forall fun _ => le_rfl)
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 153 | 154 | theorem of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.toPoly = C P.d := by |
rw [of_b_eq_zero ha hb, hc, C_0, zero_mul, zero_add]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
impor... | Mathlib/CategoryTheory/GlueData.lean | 88 | 90 | theorem t'_jii (i j : D.J) : D.t' j i i = pullback.fst ≫ D.t j i ≫ inv pullback.snd := by |
rw [← Category.assoc, ← D.t_fac]
simp
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 615 | 623 | theorem Indep.exists_insert_of_not_base (hI : M.Indep I) (hI' : ¬M.Base I) (hB : M.Base B) :
∃ e ∈ B \ I, M.Indep (insert e I) := by |
obtain ⟨B', hB', hIB'⟩ := hI.exists_base_superset
obtain ⟨x, hxB', hx⟩ := exists_of_ssubset (hIB'.ssubset_of_ne (by (rintro rfl; exact hI' hB')))
obtain (hxB | hxB) := em (x ∈ B)
· exact ⟨x, ⟨hxB, hx⟩, hB'.indep.subset (insert_subset hxB' hIB') ⟩
obtain ⟨e,he, hBase⟩ := hB'.exchange hB ⟨hxB',hxB⟩
exact ⟨e,... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 106 | 108 | theorem conjTranspose_col [Star α] (v : m → α) : (col v)ᴴ = row (star v) := by |
ext
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 Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 1,542 | 1,543 | theorem map_join (f : α → β) (L : List (List α)) : map f (join L) = join (map (map f) L) := by |
induction L <;> [rfl; simp only [*, join, map, map_append]]
|
/-
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 | 623 | 635 | theorem centroid_eq_affineCombination_of_pointsWithCircumcenter {n : ℕ} (s : Simplex ℝ P n)
(fs : Finset (Fin (n + 1))) :
fs.centroid ℝ s.points =
(univ : Finset (PointsWithCircumcenterIndex n)).affineCombination ℝ s.pointsWithCircumcenter
(centroidWeightsWithCircumcenter fs) := by |
simp_rw [centroid_def, affineCombination_apply, weightedVSubOfPoint_apply,
sum_pointsWithCircumcenter, centroidWeightsWithCircumcenter,
pointsWithCircumcenter_point, zero_smul, add_zero, centroidWeights,
← sum_indicator_subset_of_eq_zero (Function.const (Fin (n + 1)) (card fs : ℝ)⁻¹)
(fun i wi => w... |
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