Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 125 | 128 | theorem rdropWhile_last_not (hl : l.rdropWhile p ≠ []) : ¬p ((rdropWhile p l).getLast hl) := by |
simp_rw [rdropWhile]
rw [getLast_reverse]
exact dropWhile_nthLe_zero_not _ _ _
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#al... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 46 | 48 | theorem Q_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
((Q (q + 1)).f n : X _[n] ⟶ _) = (Q q).f n := by |
simp only [Q, HomologicalComplex.sub_f_apply, P_is_eventually_constant hqn]
|
/-
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.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
/-!
# ... | Mathlib/SetTheory/Cardinal/ToNat.lean | 47 | 49 | theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑(toNat c) = c := by |
lift c to ℕ using h
rw [toNat_natCast]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Subobject.MonoOver
#align_i... | Mathlib/CategoryTheory/Subterminal.lean | 107 | 110 | theorem isSubterminal_of_isIso_diag [HasBinaryProduct A A] [IsIso (diag A)] : IsSubterminal A :=
fun Z f g => by
have : (Limits.prod.fst : A ⨯ A ⟶ _) = Limits.prod.snd := by | simp [← cancel_epi (diag A)]
rw [← prod.lift_fst f g, this, prod.lift_snd]
|
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vanderm... | Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 75 | 86 | theorem det_vandermonde_id_eq_superFactorial (n : ℕ) :
(Matrix.vandermonde (fun (i : Fin (n + 1)) ↦ (i : R))).det = Nat.superFactorial n := by |
induction' n with n hn
· simp [Matrix.det_vandermonde]
· rw [Nat.superFactorial, Matrix.det_vandermonde, Fin.prod_univ_succAbove _ 0]
push_cast
congr
· simp only [Fin.val_zero, Nat.cast_zero, sub_zero]
norm_cast
simp [Fin.prod_univ_eq_prod_range (fun i ↦ (↑i + 1)) (n + 1)]
· rw [Matri... |
/-
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.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.legendre_sym... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 93 | 96 | theorem quadraticCharFun_eq_pow_of_char_ne_two (hF : ringChar F ≠ 2) {a : F} (ha : a ≠ 0) :
quadraticCharFun F a = if a ^ (Fintype.card F / 2) = 1 then 1 else -1 := by |
simp only [quadraticCharFun, ha, if_false]
simp_rw [FiniteField.isSquare_iff hF ha]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 47 | 58 | theorem length_eq_countP_add_countP (l) : length l = countP p l + countP (fun a => ¬p a) l := by |
induction l with
| nil => rfl
| cons x h ih =>
if h : p x then
rw [countP_cons_of_pos _ _ h, countP_cons_of_neg _ _ _, length, ih]
· rw [Nat.add_assoc, Nat.add_comm _ 1, Nat.add_assoc]
· simp only [h, not_true_eq_false, decide_False, not_false_eq_true]
else
rw [countP_cons_of_pos ... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.Spec... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 78 | 81 | theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by |
suffices ContDiffOn ℝ ⊤ log {0}ᶜ from this.of_le le_top
refine (contDiffOn_top_iff_deriv_of_isOpen isOpen_compl_singleton).2 ?_
simp [differentiableOn_log, contDiffOn_inv]
|
/-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 106 | 112 | theorem incMatrix_apply_eq_one_iff : G.incMatrix R a e = 1 ↔ e ∈ G.incidenceSet a := by |
-- Porting note: was `convert one_ne_zero.ite_eq_left_iff; infer_instance`
unfold incMatrix Set.indicator
simp only [Pi.one_apply]
apply Iff.intro <;> intro h
· split at h <;> simp_all only [zero_ne_one]
· simp_all only [ite_true]
|
/-
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.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c97... | Mathlib/RingTheory/WittVector/Truncated.lean | 145 | 146 | theorem coeff_truncateFun (x : 𝕎 R) (i : Fin n) : (truncateFun n x).coeff i = x.coeff i := by |
rw [truncateFun, TruncatedWittVector.coeff_mk]
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Order.Partition.Equipartition
#align_... | Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 111 | 112 | theorem m_pos [Nonempty α] (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α) : 0 < m := by |
sz_positivity
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 69 | 74 | theorem le_exp_log (x : ℝ) : x ≤ exp (log x) := by |
by_cases h_zero : x = 0
· rw [h_zero, log, dif_pos rfl, exp_zero]
exact zero_le_one
· rw [exp_log_eq_abs h_zero]
exact le_abs_self _
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"90... | Mathlib/RingTheory/Complex.lean | 37 | 40 | theorem Algebra.norm_complex_apply (z : ℂ) : Algebra.norm ℝ z = Complex.normSq z := by |
rw [Algebra.norm_eq_matrix_det Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.det_fin_two, normSq_apply]
simp
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 63 | 68 | theorem convexBodyLT_mem {x : K} :
mixedEmbedding K x ∈ (convexBodyLT K f) ↔ ∀ w : InfinitePlace K, w x < f w := by |
simp_rw [mixedEmbedding, RingHom.prod_apply, Set.mem_prod, Set.mem_pi, Set.mem_univ,
forall_true_left, mem_ball_zero_iff, Pi.ringHom_apply, ← Complex.norm_real,
embedding_of_isReal_apply, Subtype.forall, ← forall₂_or_left, ← not_isReal_iff_isComplex, em,
forall_true_left, norm_embedding_eq]
|
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Lie.Basic
#align_import algebr... | Mathlib/Algebra/Lie/DirectSum.lean | 130 | 136 | theorem lie_of_of_ne [DecidableEq ι] {i j : ι} (hij : i ≠ j) (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = 0 := by |
refine DFinsupp.ext fun k => ?_
rw [bracket_apply]
obtain rfl | hik := Decidable.eq_or_ne i k
· rw [of_eq_of_ne _ _ _ _ hij.symm, lie_zero, zero_apply]
· rw [of_eq_of_ne _ _ _ _ hik, zero_lie, zero_apply]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 106 | 113 | theorem BlankExtends.above_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
BlankExtends l₁ l → BlankExtends l₂ l → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by |
rintro ⟨i, rfl⟩ ⟨j, e⟩ h; use i - j
refine List.append_cancel_right (e.symm.trans ?_)
rw [List.append_assoc, ← List.replicate_add, Nat.sub_add_cancel]
apply_fun List.length at e
simp only [List.length_append, List.length_replicate] at e
rwa [← Nat.add_le_add_iff_left, e, Nat.add_le_add_iff_right]
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Rin... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 116 | 122 | theorem floor_eq_self_of_mem (m : E) (h : m ∈ span ℤ (Set.range b)) : (floor b m : E) = m := by |
apply b.ext_elem
simp_rw [repr_floor_apply b]
intro i
obtain ⟨z, hz⟩ := (b.mem_span_iff_repr_mem ℤ _).mp h i
rw [← hz]
exact congr_arg (Int.cast : ℤ → K) (Int.floor_intCast z)
|
/-
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.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320... | Mathlib/Order/Bounds/OrderIso.lean | 59 | 60 | theorem isLUB_preimage' {s : Set β} {x : β} : IsLUB (f ⁻¹' s) (f.symm x) ↔ IsLUB s x := by |
rw [isLUB_preimage, f.apply_symm_apply]
|
/-
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.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.categor... | Mathlib/CategoryTheory/Functor/Category.lean | 68 | 68 | theorem congr_app {α β : F ⟶ G} (h : α = β) (X : C) : α.app X = β.app X := by | rw [h]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 112 | 116 | theorem imageToKernel_comp_right {D : V} (h : C ⟶ D) (w : f ≫ g = 0) :
imageToKernel f (g ≫ h) (by simp [reassoc_of% w]) =
imageToKernel f g w ≫ Subobject.ofLE _ _ (kernelSubobject_comp_le g h) := by |
ext
simp
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 17 | 18 | theorem swap_then (o₁ o₂ : Ordering) : (o₁.then o₂).swap = o₁.swap.then o₂.swap := by |
cases o₁ <;> rfl
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.inv from "leanpro... | Mathlib/Analysis/Calculus/Deriv/Inv.lean | 87 | 90 | theorem deriv_inv : deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹ := by |
rcases eq_or_ne x 0 with (rfl | hne)
· simp [deriv_zero_of_not_differentiableAt (mt differentiableAt_inv.1 (not_not.2 rfl))]
· exact (hasDerivAt_inv hne).deriv
|
/-
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 | 40 | 44 | theorem coeff_add (p q : R[X]) (n : ℕ) : coeff (p + q) n = coeff p n + coeff q n := by |
rcases p with ⟨⟩
rcases q with ⟨⟩
simp_rw [← ofFinsupp_add, coeff]
exact Finsupp.add_apply _ _ _
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-com... | Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 23 | 26 | theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :
(∏ p ∈ antidiagonal (n + 1), f p)
= f (0, n + 1) * ∏ p ∈ antidiagonal n, f (p.1 + 1, p.2) := by |
rw [antidiagonal_succ, prod_cons, prod_map]; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
/-!
# Infinite sums and products over `ℕ` and `ℤ`
This file contains lemma... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 68 | 70 | theorem zero_mul {f : ℕ → M} (h : HasProd (fun n ↦ f (n + 1)) m) :
HasProd f (f 0 * m) := by |
simpa only [prod_range_one] using h.prod_range_mul
|
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.nat.factorial.big_operators from "leanprover-commun... | Mathlib/Data/Nat/Factorial/BigOperators.lean | 31 | 31 | theorem prod_factorial_pos : 0 < ∏ i ∈ s, (f i)! := by | positivity
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Simon Hudon
-/
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.lis... | Mathlib/Data/List/TFAE.lean | 56 | 57 | theorem tfae_cons_self {a} {l : List Prop} : TFAE (a :: a :: l) ↔ TFAE (a :: l) := by |
simp [tfae_cons_cons]
|
/-
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 | 105 | 113 | theorem condexpIndL1Fin_smul (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : ℝ) (x : G) :
condexpIndL1Fin hm hs hμs (c • x) = c • condexpIndL1Fin hm hs hμs x := by |
ext1
refine (Memℒp.coeFn_toLp q).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_smul _ _).symm
rw [condexpIndSMul_smul hs hμs c x]
refine (Lp.coeFn_smul _ _).trans ?_
refine (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun y hy => ?_
simp only [Pi.smul_apply, hy]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Algebra.Ring.Int
import Mathlib.Al... | Mathlib/Tactic/NormNum/Basic.lean | 104 | 105 | theorem isNat_natCast {R} [AddMonoidWithOne R] (n m : ℕ) :
IsNat n m → IsNat (n : R) m := by | rintro ⟨⟨⟩⟩; exact ⟨rfl⟩
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
imp... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 63 | 98 | theorem exists_approx_polynomial_aux [Ring Fq] {d : ℕ} {m : ℕ} (hm : Fintype.card Fq ^ d ≤ m)
(b : Fq[X]) (A : Fin m.succ → Fq[X]) (hA : ∀ i, degree (A i) < degree b) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d) := by |
have hb : b ≠ 0 := by
rintro rfl
specialize hA 0
rw [degree_zero] at hA
exact not_lt_of_le bot_le hA
-- Since there are > q^d elements of A, and only q^d choices for the highest `d` coefficients,
-- there must be two elements of A with the same coefficients at
-- `degree b - 1`, ... `degree b -... |
/-
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.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Su... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 76 | 78 | theorem flip (n ν : ℕ) (h : ν ≤ n) :
(bernsteinPolynomial R n ν).comp (1 - X) = bernsteinPolynomial R n (n - ν) := by |
simp [bernsteinPolynomial, h, tsub_tsub_assoc, mul_right_comm]
|
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 120 | 125 | theorem AffineIndependent.finrank_vectorSpan [Fintype ι] {p : ι → P} (hi : AffineIndependent k p)
{n : ℕ} (hc : Fintype.card ι = n + 1) : finrank k (vectorSpan k (Set.range p)) = n := by |
classical
rw [← Finset.card_univ] at hc
rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image]
exact hi.finrank_vectorSpan_image_finset hc
|
/-
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.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.double_counting from "leanprover-community/mathli... | Mathlib/Combinatorics/Enumerative/DoubleCounting.lean | 79 | 82 | theorem sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow [∀ a b, Decidable (r a b)] :
(∑ a ∈ s, (t.bipartiteAbove r a).card) = ∑ b ∈ t, (s.bipartiteBelow r b).card := by |
simp_rw [card_eq_sum_ones, bipartiteAbove, bipartiteBelow, sum_filter]
exact sum_comm
|
/-
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.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multis... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 91 | 94 | theorem prod_map_erase [DecidableEq ι] {a : ι} (h : a ∈ m) :
f a * ((m.erase a).map f).prod = (m.map f).prod := by |
rw [← m.coe_toList, coe_erase, map_coe, map_coe, prod_coe, prod_coe,
List.prod_map_erase f (mem_toList.2 h)]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 83 | 88 | theorem bernoulli'_spec (n : ℕ) :
(∑ k ∈ range n.succ, (n.choose (n - k) : ℚ) / (n - k + 1) * bernoulli' k) = 1 := by |
rw [sum_range_succ_comm, bernoulli'_def n, tsub_self, choose_zero_right, sub_self, zero_add,
div_one, cast_one, one_mul, sub_add, ← sum_sub_distrib, ← sub_eq_zero, sub_sub_cancel_left,
neg_eq_zero]
exact Finset.sum_eq_zero (fun x hx => by rw [choose_symm (le_of_lt (mem_range.1 hx)), sub_self])
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 82 | 83 | theorem separable_C (r : R) : (C r).Separable ↔ IsUnit r := by |
rw [separable_def, derivative_C, isCoprime_zero_right, isUnit_C]
|
/-
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.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Operations
/-!
# The colon ideal
This file defines `Submodule.colon N P` as the ideal of all elements `r... | Mathlib/RingTheory/Ideal/Colon.lean | 86 | 87 | theorem _root_.Ideal.annihilator_quotient {I : Ideal R} : Module.annihilator R (R ⧸ I) = I := by |
rw [Submodule.annihilator_quotient, colon_top]
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprove... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 102 | 113 | theorem LDL.diag_eq_lowerInv_conj : LDL.diag hS = LDL.lowerInv hS * S * (LDL.lowerInv hS)ᴴ := by |
ext i j
by_cases hij : i = j
· simp only [diag, diagEntries, EuclideanSpace.inner_piLp_equiv_symm, star_star, hij,
diagonal_apply_eq, Matrix.mul_assoc]
rfl
· simp only [LDL.diag, hij, diagonal_apply_ne, Ne, not_false_iff, mul_mul_apply]
rw [conjTranspose, transpose_map, transpose_transpose, dotProd... |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
... | Mathlib/Data/Nat/Dist.lean | 57 | 57 | theorem dist_tri_right (n m : ℕ) : m ≤ n + dist n m := by | rw [add_comm]; apply dist_tri_left
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Mario Carneiro, Martin Dvorak
-/
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac... | Mathlib/Data/List/Join.lean | 60 | 62 | theorem join_filter_ne_nil [DecidablePred fun l : List α => l ≠ []] {L : List (List α)} :
join (L.filter fun l => l ≠ []) = L.join := by |
simp [join_filter_not_isEmpty, ← isEmpty_iff_eq_nil]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring
import Mathlib... | Mathlib/Data/Nat/Choose/Sum.lean | 37 | 67 | theorem add_pow (h : Commute x y) (n : ℕ) :
(x + y) ^ n = ∑ m ∈ range (n + 1), x ^ m * y ^ (n - m) * choose n m := by |
let t : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * choose n m
change (x + y) ^ n = ∑ m ∈ range (n + 1), t n m
have h_first : ∀ n, t n 0 = y ^ n := fun n ↦ by
simp only [t, choose_zero_right, _root_.pow_zero, Nat.cast_one, mul_one, one_mul, tsub_zero]
have h_last : ∀ n, t n n.succ = 0 := fun n ↦ by
si... |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
/-!
# The structure... | Mathlib/Data/Fintype/Fin.lean | 41 | 51 | theorem Ioi_succ (i : Fin n) : Ioi i.succ = (Ioi i).map (Fin.succEmb _) := by |
ext i
simp only [mem_filter, mem_Ioi, mem_map, mem_univ, true_and_iff, Function.Embedding.coeFn_mk,
exists_true_left]
constructor
· refine cases ?_ ?_ i
· rintro ⟨⟨⟩⟩
· intro i hi
exact ⟨i, succ_lt_succ_iff.mp hi, rfl⟩
· rintro ⟨i, hi, rfl⟩
simpa
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 131 | 133 | theorem rdropWhile_prefix : l.rdropWhile p <+: l := by |
rw [← reverse_suffix, rdropWhile, reverse_reverse]
exact dropWhile_suffix _
|
/-
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 | 108 | 110 | theorem injective_boxes : Function.Injective (boxes : Prepartition I → Finset (Box ι)) := by |
rintro ⟨s₁, h₁, h₁'⟩ ⟨s₂, h₂, h₂'⟩ (rfl : s₁ = s₂)
rfl
|
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.h... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 72 | 74 | theorem hermite_one : hermite 1 = X := by |
rw [hermite_succ, hermite_zero]
simp only [map_one, mul_one, derivative_one, sub_zero]
|
/-
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.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ri... | Mathlib/RingTheory/WittVector/Basic.lean | 105 | 105 | theorem one : mapFun f (1 : 𝕎 R) = 1 := by | map_fun_tac
|
/-
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 | 70 | 71 | theorem inv_subst {α} [Field α] {n k e : α} (h2 : e ≠ 0) (h3 : n * e = k) :
k * (e ⁻¹) = n := by | rw [← div_eq_mul_inv, ← h3, mul_div_cancel_right₀ _ h2]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_localizati... | Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 109 | 115 | theorem ext {c1 c2 : NumDenSameDeg 𝒜 x} (hdeg : c1.deg = c2.deg) (hnum : (c1.num : A) = c2.num)
(hden : (c1.den : A) = c2.den) : c1 = c2 := by |
rcases c1 with ⟨i1, ⟨n1, hn1⟩, ⟨d1, hd1⟩, h1⟩
rcases c2 with ⟨i2, ⟨n2, hn2⟩, ⟨d2, hd2⟩, h2⟩
dsimp only [Subtype.coe_mk] at *
subst hdeg hnum hden
congr
|
/-
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.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
#align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 99 | 107 | theorem intersecting_iff_eq_empty_of_subsingleton [Subsingleton α] (s : Set α) :
s.Intersecting ↔ s = ∅ := by |
refine
subsingleton_of_subsingleton.intersecting.trans
⟨not_imp_comm.2 fun h => subsingleton_of_subsingleton.eq_singleton_of_mem ?_, ?_⟩
· obtain ⟨a, ha⟩ := nonempty_iff_ne_empty.2 h
rwa [Subsingleton.elim ⊥ a]
· rintro rfl
exact (Set.singleton_nonempty _).ne_empty.symm
|
/-
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.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 57 | 83 | theorem quasiSeparatedSpace_iff_affine (X : Scheme) :
QuasiSeparatedSpace X.carrier ↔ ∀ U V : X.affineOpens, IsCompact (U ∩ V : Set X.carrier) := by |
rw [quasiSeparatedSpace_iff]
constructor
· intro H U V; exact H U V U.1.2 U.2.isCompact V.1.2 V.2.isCompact
· intro H
suffices
∀ (U : Opens X.carrier) (_ : IsCompact U.1) (V : Opens X.carrier) (_ : IsCompact V.1),
IsCompact (U ⊓ V).1
by intro U V hU hU' hV hV'; exact this ⟨U, hU⟩ hU' ⟨V... |
/-
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 | 108 | 113 | theorem Perm.length_eq {l₁ l₂ : List α} (p : l₁ ~ l₂) : length l₁ = length l₂ := by |
induction p with
| nil => rfl
| cons _ _ ih => simp only [length_cons, ih]
| swap => rfl
| trans _ _ ih₁ ih₂ => simp only [ih₁, ih₂]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Valuation.Basic
#align_import ring_theory.valuation.integers from "leanprover-community/mathlib"@"7b7da89322fe46a16bf03eeb345b0acfc73fe10e"
/-!
# Ri... | Mathlib/RingTheory/Valuation/Integers.lean | 83 | 93 | theorem isUnit_of_one {x : O} (hx : IsUnit (algebraMap O R x)) (hvx : v (algebraMap O R x) = 1) :
IsUnit x :=
let ⟨u, hu⟩ := hx
have h1 : v u ≤ 1 := hu.symm ▸ hv.2 x
have h2 : v (u⁻¹ : Rˣ) ≤ 1 := by |
rw [← one_mul (v _), ← hvx, ← v.map_mul, ← hu, u.mul_inv, hu, hvx, v.map_one]
let ⟨r1, hr1⟩ := hv.3 h1
let ⟨r2, hr2⟩ := hv.3 h2
⟨⟨r1, r2, hv.1 <| by rw [RingHom.map_mul, RingHom.map_one, hr1, hr2, Units.mul_inv],
hv.1 <| by rw [RingHom.map_mul, RingHom.map_one, hr1, hr2, Units.inv_mul]⟩,
hv.1 <| hr... |
/-
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.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 126 | 129 | theorem nonMemberSubfamily_memberSubfamily :
(𝒜.memberSubfamily a).nonMemberSubfamily a = 𝒜.memberSubfamily a := by |
ext
simp
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 120 | 122 | theorem edgeFinset_top [DecidableEq V] :
(⊤ : SimpleGraph V).edgeFinset = univ.filter fun e => ¬e.IsDiag := by |
rw [← coe_inj]; simp
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
#alig... | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 79 | 95 | theorem succNthDefiningPoly_degree [IsDomain k] (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : Fin (n + 1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
(succNthDefiningPoly p n a₁ a₂ bs).degree = p := by |
have : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1))).degree = (p : WithBot ℕ) := by
rw [degree_mul, degree_C]
· simp only [Nat.cast_withBot, add_zero, degree_X, degree_pow, Nat.smul_one_eq_cast]
· exact pow_ne_zero _ ha₁
have : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1)) - X * C (a₂.coeff 0 ^ p ^ (n + 1))).degree =... |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated
import Mathlib.AlgebraicGeometry.Pullbacks
im... | Mathlib/AlgebraicGeometry/Morphisms/Separated.lean | 49 | 52 | theorem isSeparated_eq_diagonal_isClosedImmersion :
@IsSeparated = MorphismProperty.diagonal @IsClosedImmersion := by |
ext
exact isSeparated_iff _
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!... | Mathlib/Topology/NhdsSet.lean | 35 | 38 | theorem nhdsSet_diagonal (X) [TopologicalSpace (X × X)] :
𝓝ˢ (diagonal X) = ⨆ (x : X), 𝓝 (x, x) := by |
rw [nhdsSet, ← range_diag, ← range_comp]
rfl
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-communit... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 95 | 111 | theorem GammaIntegral_convergent {s : ℂ} (hs : 0 < s.re) :
IntegrableOn (fun x => (-x).exp * x ^ (s - 1) : ℝ → ℂ) (Ioi 0) := by |
constructor
· refine ContinuousOn.aestronglyMeasurable ?_ measurableSet_Ioi
apply (continuous_ofReal.comp continuous_neg.rexp).continuousOn.mul
apply ContinuousAt.continuousOn
intro x hx
have : ContinuousAt (fun x : ℂ => x ^ (s - 1)) ↑x :=
continuousAt_cpow_const <| ofReal_mem_slitPlane.2 hx
... |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
#align_import ring_... | Mathlib/RingTheory/Derivation/Basic.lean | 147 | 149 | theorem map_algebraMap : D (algebraMap R A r) = 0 := by |
rw [← mul_one r, RingHom.map_mul, RingHom.map_one, ← smul_def, map_smul, map_one_eq_zero,
smul_zero]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | Mathlib/Topology/Inseparable.lean | 50 | 75 | theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by |
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Dual
import Mathlib.Data.Fin.FlagRange
/-!
# Flag of submodules defined by a basis
... | Mathlib/LinearAlgebra/Basis/Flag.lean | 83 | 86 | theorem flag_le_ker_dual (b : Basis (Fin n) R M) (k : Fin n) :
b.flag k.castSucc ≤ LinearMap.ker (b.dualBasis k) := by |
nontriviality R
rw [coe_dualBasis, b.flag_le_ker_coord_iff]
|
/-
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.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#ali... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 67 | 70 | theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v).card = G.degree v := by |
simpa only [dart_fst_fiber, Finset.card_univ, card_neighborSet_eq_degree] using
card_image_of_injective univ (G.dartOfNeighborSet_injective v)
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 89 | 96 | theorem exists_pow_nat_eq [IsAlgClosed k] (x : k) {n : ℕ} (hn : 0 < n) : ∃ z, z ^ n = x := by |
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn x]
exact ne_of_gt (WithBot.coe_lt_coe.2 hn)
obtain ⟨z, hz⟩ := exists_root (X ^ n - C x) this
use z
simp only [eval_C, eval_X, eval_pow, eval_sub, IsRoot.def] at hz
exact sub_eq_zero.1 hz
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 125 | 127 | theorem map_rootsOfUnity (f : Mˣ →* Nˣ) (k : ℕ+) : (rootsOfUnity k M).map f ≤ rootsOfUnity k N := by |
rintro _ ⟨ζ, h, rfl⟩
simp_all only [← map_pow, mem_rootsOfUnity, SetLike.mem_coe, MonoidHom.map_one]
|
/-
Copyright (c) 2020 Google LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Wong
-/
import Mathlib.Data.List.Basic
#align_import data.list.palindrome from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Palindromes
This mod... | Mathlib/Data/List/Palindrome.lean | 55 | 61 | theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by |
refine bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) ?_
intro x l y hp hr
rw [reverse_cons, reverse_append] at hr
rw [head_eq_of_cons_eq hr]
have : Palindrome l := hp (append_inj_left' (tail_eq_of_cons_eq hr) rfl)
exact Palindrome.cons_concat x this
|
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 100 | 105 | theorem count_roots [DecidableEq R] (p : R[X]) : p.roots.count a = rootMultiplicity a p := by |
classical
by_cases hp : p = 0
· simp [hp]
rw [roots_def, dif_neg hp]
exact (Classical.choose_spec (exists_multiset_roots hp)).2 a
|
/-
Copyright (c) 2019 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.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6e... | Mathlib/Data/Rat/Cast/Lemmas.lean | 28 | 32 | theorem cast_inv_nat (n : ℕ) : ((n⁻¹ : ℚ) : α) = (n : α)⁻¹ := by |
cases' n with n
· simp
rw [cast_def, inv_natCast_num, inv_natCast_den, if_neg n.succ_ne_zero,
Int.sign_eq_one_of_pos (Nat.cast_pos.mpr n.succ_pos), Int.cast_one, one_div]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | Mathlib/Order/OrderIsoNat.lean | 99 | 101 | theorem not_wellFounded_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) : ¬WellFounded r := by |
rw [wellFounded_iff_no_descending_seq, not_isEmpty_iff]
exact ⟨f⟩
|
/-
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 | 83 | 91 | theorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1 := by |
induction a using limitRecOn with
| H₁ => simp only [opow_zero]
| H₂ _ ih =>
simp only [opow_succ, ih, mul_one]
| H₃ b l IH =>
refine eq_of_forall_ge_iff fun c => ?_
rw [opow_le_of_limit Ordinal.one_ne_zero l]
exact ⟨fun H => by simpa only [opow_zero] using H 0 l.pos, fun H b' h => by rwa [IH _... |
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
import Mathlib.... | Mathlib/Topology/VectorBundle/Constructions.lean | 96 | 106 | theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChangeL 𝕜 e₂... |
rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap']
rintro ⟨v₁, v₂⟩
show
(e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) =
(e₁.coordChangeL 𝕜 e₁' b v₁, e₂.coordChangeL 𝕜 e₂' b v₂)
rw [e₁.coordChangeL_apply e₁', e₂.coordChangeL_apply e₂', (e₁.prod e₂).coordChangeL_apply']
exa... |
/-
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 | 149 | 150 | theorem symmDiff_le_iff {a b c : α} : a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c := by |
simp_rw [symmDiff, sup_le_iff, sdiff_le_iff]
|
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Antoine Labelle
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.Tens... | Mathlib/Algebra/Module/Projective.lean | 92 | 94 | theorem projective_def' :
Projective R P ↔ ∃ s : P →ₗ[R] P →₀ R, Finsupp.total P P R id ∘ₗ s = .id := by |
simp_rw [projective_def, DFunLike.ext_iff, Function.LeftInverse, comp_apply, id_apply]
|
/-
Copyright (c) 2021 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak, Kyle Miller, Eric Wieser
-/
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Math... | Mathlib/LinearAlgebra/CrossProduct.lean | 103 | 106 | theorem dot_self_cross (v w : Fin 3 → R) : v ⬝ᵥ v ×₃ w = 0 := by |
rw [cross_apply, vec3_dotProduct]
set_option tactic.skipAssignedInstances false in norm_num
ring
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Lo... | Mathlib/RingTheory/Localization/Module.lean | 73 | 76 | theorem LinearIndependent.localization {ι : Type*} {b : ι → M} (hli : LinearIndependent R b) :
LinearIndependent Rₛ b := by |
have := isLocalizedModule_id S M Rₛ
exact hli.of_isLocalizedModule Rₛ S .id
|
/-
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.RingTheory.LocalProperties
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import ring_theory.ring_hom.finite_type from "leanprover-community/mat... | Mathlib/RingTheory/RingHom/FiniteType.lean | 38 | 91 | theorem finiteType_ofLocalizationSpanTarget : OfLocalizationSpanTarget @FiniteType := by |
-- Setup algebra intances.
rw [ofLocalizationSpanTarget_iff_finite]
introv R hs H
classical
letI := f.toAlgebra
replace H : ∀ r : s, Algebra.FiniteType R (Localization.Away (r : S)) := by
intro r; simp_rw [RingHom.FiniteType] at H; convert H r; ext; simp_rw [Algebra.smul_def]; rfl
replace H := fun r ... |
/-
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.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
#align_import category_theory.natural... | Mathlib/CategoryTheory/NatIso.lean | 131 | 132 | theorem cancel_natIso_inv_left {X : C} {Z : D} (g g' : F.obj X ⟶ Z) :
α.inv.app X ≫ g = α.inv.app X ≫ g' ↔ g = g' := by | simp only [cancel_epi, refl]
|
/-
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.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 120 | 121 | theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by |
rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | Mathlib/Algebra/BigOperators/Group/List.lean | 150 | 151 | theorem prod_eq_pow_card (l : List M) (m : M) (h : ∀ x ∈ l, x = m) : l.prod = m ^ l.length := by |
rw [← prod_replicate, ← List.eq_replicate.mpr ⟨rfl, h⟩]
|
/-
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.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Bilin... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 93 | 103 | theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι]
[DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
LinearMap.range ((Algebra.linearMap C L).restrictScalars A) ≤
Submodule.span A (Set.range <| (traceForm K L).dualBasis (tr... |
rw [← LinearMap.BilinForm.dualSubmodule_span_of_basis,
← LinearMap.BilinForm.le_flip_dualSubmodule, Submodule.span_le]
rintro _ ⟨i, rfl⟩ _ ⟨y, rfl⟩
simp only [LinearMap.coe_restrictScalars, linearMap_apply, LinearMap.BilinForm.flip_apply,
traceForm_apply]
refine IsIntegrallyClosed.isIntegral_iff.mp ?_
... |
/-
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.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multis... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 85 | 86 | theorem prod_erase [DecidableEq α] (h : a ∈ s) : a * (s.erase a).prod = s.prod := by |
rw [← s.coe_toList, coe_erase, prod_coe, prod_coe, List.prod_erase (mem_toList.2 h)]
|
/-
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.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
/-!
# Integration with respect to a finite product of measures
... | Mathlib/MeasureTheory/Integral/Pi.lean | 26 | 41 | theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
{f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) := by |
induction n with
| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,
integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]
| succ n n_ih =>
have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)
rw [volu... |
/-
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.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 40 | 57 | theorem derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors
{p : R[X]} {t : R} (hpt : Polynomial.IsRoot p t)
(hnzd : (p.rootMultiplicity t : R) ∈ nonZeroDivisors R) :
(derivative p).rootMultiplicity t = p.rootMultiplicity t - 1 := by |
by_cases h : p = 0
· simp only [h, map_zero, rootMultiplicity_zero]
obtain ⟨g, hp, hndvd⟩ := p.exists_eq_pow_rootMultiplicity_mul_and_not_dvd h t
set m := p.rootMultiplicity t
have hm : m - 1 + 1 = m := Nat.sub_add_cancel <| (rootMultiplicity_pos h).2 hpt
have hndvd : ¬(X - C t) ^ m ∣ derivative p := by
... |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 109 | 111 | theorem mem_roots' : a ∈ p.roots ↔ p ≠ 0 ∧ IsRoot p a := by |
classical
rw [← count_pos, count_roots p, rootMultiplicity_pos']
|
/-
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.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 109 | 110 | theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by | rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
|
/-
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 | 78 | 87 | theorem Ideal.radical_minimalPrimes : I.radical.minimalPrimes = I.minimalPrimes := by |
rw [Ideal.minimalPrimes, Ideal.minimalPrimes]
ext p
refine ⟨?_, ?_⟩ <;> rintro ⟨⟨a, ha⟩, b⟩
· refine ⟨⟨a, a.radical_le_iff.1 ha⟩, ?_⟩
simp only [Set.mem_setOf_eq, and_imp] at *
exact fun _ h2 h3 h4 => b h2 (h2.radical_le_iff.2 h3) h4
· refine ⟨⟨a, a.radical_le_iff.2 ha⟩, ?_⟩
simp only [Set.mem_se... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 154 | 155 | theorem content_monomial {r : R} {k : ℕ} : content (monomial k r) = normalize r := by |
rw [← C_mul_X_pow_eq_monomial, content_C_mul, content_X_pow, mul_one]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e... | Mathlib/CategoryTheory/Subobject/Limits.lean | 134 | 137 | theorem factorThruKernelSubobject_comp_arrow {W : C} (h : W ⟶ X) (w : h ≫ f = 0) :
factorThruKernelSubobject f h w ≫ (kernelSubobject f).arrow = h := by |
dsimp [factorThruKernelSubobject]
simp
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathl... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 55 | 63 | theorem exists_mem_of_not_isUnit_aeval_prod [IsDomain R] {p : R[X]} {a : A}
(h : ¬IsUnit (aeval a (Multiset.map (fun x : R => X - C x) p.roots).prod)) :
∃ k : R, k ∈ σ a ∧ eval k p = 0 := by |
rw [← Multiset.prod_toList, AlgHom.map_list_prod] at h
replace h := mt List.prod_isUnit h
simp only [not_forall, exists_prop, aeval_C, Multiset.mem_toList, List.mem_map, aeval_X,
exists_exists_and_eq_and, Multiset.mem_map, AlgHom.map_sub] at h
rcases h with ⟨r, r_mem, r_nu⟩
exact ⟨r, by rwa [mem_iff, ← I... |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 91 | 93 | theorem snorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by |
rw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn, Mario Carneiro
-/
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f :... | .lake/packages/batteries/Batteries/Logic.lean | 88 | 91 | theorem eqRec_eq_cast {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') :
@Eq.rec α a motive x a' e = cast (e ▸ rfl) x := by |
subst e; rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
#align_import algebra.group_with_zero.basic from "leanprov... | Mathlib/Algebra/GroupWithZero/Basic.lean | 110 | 111 | theorem eq_zero_of_zero_eq_one (h : (0 : M₀) = 1) (a : M₀) : a = 0 := by |
rw [← mul_one a, ← h, mul_zero]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 87 | 90 | theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by |
constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
simp only [mulLeft_eq_zero_iff, pow_mulLeft] at hn ⊢ <;>
exact hn
|
/-
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 | 98 | 99 | theorem mem_vars (i : σ) : i ∈ p.vars ↔ ∃ d ∈ p.support, i ∈ d.support := by |
classical simp only [vars_def, Multiset.mem_toFinset, mem_degrees, mem_support_iff, exists_prop]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
impor... | Mathlib/LinearAlgebra/Vandermonde.lean | 72 | 74 | theorem vandermonde_transpose_mul_vandermonde {n : ℕ} (v : Fin n → R) (i j) :
((vandermonde v)ᵀ * vandermonde v) i j = ∑ k : Fin n, v k ^ (i + j : ℕ) := by |
simp only [vandermonde_apply, Matrix.mul_apply, Matrix.transpose_apply, pow_add]
|
/-
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 | 99 | 103 | theorem cycleOf_apply_apply_zpow_self (f : Perm α) (x : α) (k : ℤ) :
cycleOf f x ((f ^ k) x) = (f ^ (k + 1) : Perm α) x := by |
rw [SameCycle.cycleOf_apply]
· rw [add_comm, zpow_add, zpow_one, mul_apply]
· exact ⟨k, rfl⟩
|
/-
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.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 115 | 116 | theorem derivative_X_sq : derivative (X ^ 2 : R[X]) = C 2 * X := by |
rw [derivative_X_pow, Nat.cast_two, pow_one]
|
/-
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.Data.Multiset.Nodup
#align_import data.multiset.sum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Disjoint sum of ... | Mathlib/Data/Multiset/Sum.lean | 55 | 60 | theorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s := by |
rw [mem_disjSum, or_iff_left]
-- Porting note: Previous code for L62 was: simp only [exists_eq_right]
· simp only [inl.injEq, exists_eq_right]
rintro ⟨b, _, hb⟩
exact inr_ne_inl hb
|
/-
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.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c97... | Mathlib/RingTheory/WittVector/Truncated.lean | 166 | 167 | theorem truncateFun_out (x : TruncatedWittVector p n R) : x.out.truncateFun n = x := by |
simp only [WittVector.truncateFun, coeff_out, mk_coeff]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.SetTheory.Cardinal.Basic
#align_import data.rat.denumerable from "leanprover-community/mathlib"@"dde670c9a3f503647fd... | Mathlib/Data/Rat/Denumerable.lean | 45 | 45 | theorem Cardinal.mkRat : #ℚ = ℵ₀ := by | simp only [mk_eq_aleph0]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
/-!
# Subsingleton
Defines the predicate `Subsingleton s : Prop`, saying that `s` has at most one element.
Also defi... | Mathlib/Data/Set/Subsingleton.lean | 99 | 104 | theorem exists_eq_singleton_iff_nonempty_subsingleton :
(∃ a : α, s = {a}) ↔ s.Nonempty ∧ s.Subsingleton := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨a, rfl⟩
exact ⟨singleton_nonempty a, subsingleton_singleton⟩
· exact h.2.eq_empty_or_singleton.resolve_left h.1.ne_empty
|
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