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) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fi... | Mathlib/Data/Fin/Tuple/Sort.lean | 50 | 57 | theorem graph.card (f : Fin n → α) : (graph f).card = n := by |
rw [graph, Finset.card_image_of_injective]
· exact Finset.card_fin _
· intro _ _
-- porting note (#10745): was `simp`
dsimp only
rw [Prod.ext_iff]
simp
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Compl... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 121 | 124 | theorem snd : IsBoundedLinearMap 𝕜 fun x : E × F => x.2 := by |
refine (LinearMap.snd 𝕜 E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_right _ _
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import algebra.category.Module.images from "leanprover-co... | Mathlib/Algebra/Category/ModuleCat/Images.lean | 117 | 119 | theorem imageIsoRange_hom_subtype {G H : ModuleCat.{v} R} (f : G ⟶ H) :
(imageIsoRange f).hom ≫ ModuleCat.ofHom f.range.subtype = Limits.image.ι f := by |
erw [← imageIsoRange_inv_image_ι f, Iso.hom_inv_id_assoc]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 95 | 97 | theorem support_formPerm_le' : { x | formPerm l x ≠ x } ≤ l.toFinset := by |
refine (zipWith_swap_prod_support' l l.tail).trans ?_
simpa [Finset.subset_iff] using tail_subset l
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/math... | Mathlib/Data/Complex/Cardinality.lean | 31 | 31 | theorem mk_univ_complex : #(Set.univ : Set ℂ) = 𝔠 := by | rw [mk_univ, mk_complex]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 72 | 75 | theorem finrank_le_of_rank_le {n : ℕ} (h : Module.rank R M ≤ ↑n) : finrank R M ≤ n := by |
rwa [← Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_natCast] at h
· exact h.trans_lt (nat_lt_aleph0 n)
· exact nat_lt_aleph0 n
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 150 | 151 | theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by |
erw [coeff, ← h, ← Finsupp.unique_single s]
|
/-
Copyright (c) 2022 Siddhartha Prasad, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Siddhartha Prasad, Yaël Dillies
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Tactic.Monotonici... | Mathlib/Algebra/Order/Kleene.lean | 154 | 154 | theorem add_eq_right_iff_le : a + b = b ↔ a ≤ b := by | simp
|
/-
Copyright (c) 2023 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mat... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 110 | 112 | theorem toPrefunctor_symm_comp (f : SingleObj α ⥤q SingleObj β) (g : SingleObj β ⥤q SingleObj γ) :
toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f := by |
simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]
|
/-
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.List.Join
#align_import data.list.permutation from "leanprover-community/... | Mathlib/Data/List/Permutation.lean | 77 | 79 | theorem permutationsAux2_append (t : α) (ts : List α) (r : List β) (ys : List α) (f : List α → β) :
(permutationsAux2 t ts nil ys f).2 ++ r = (permutationsAux2 t ts r ys f).2 := by |
induction ys generalizing f <;> simp [*]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 67 | 68 | theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
0 ≤ re ⟪x, T x⟫ := by | rw [inner_re_symm]; exact hT.inner_nonneg_left x
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | Mathlib/Algebra/Category/ModuleCat/Abelian.lean | 123 | 127 | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g := by |
rw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
exact
⟨fun h => le_antisymm (range_le_ker_iff.2 h.1) (ker_le_range_iff.2 h.2), fun h =>
⟨range_le_ker_iff.1 <| le_of_eq h, ker_le_range_iff.1 <| le_of_eq h.symm⟩⟩
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analy... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 62 | 65 | theorem le_rpow_one_add_norm_iff_norm_le {r t : ℝ} (hr : 0 < r) (ht : 0 < t) (x : E) :
t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1 := by |
rw [le_sub_iff_add_le', neg_inv]
exact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm
|
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 36 | 37 | theorem eval₂_one_cyclotomic_prime {R S : Type*} [CommRing R] [Semiring S] (f : R →+* S) {p : ℕ}
[Fact p.Prime] : eval₂ f 1 (cyclotomic p R) = p := by | simp
|
/-
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.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"... | Mathlib/Data/Rat/Cast/CharZero.lean | 46 | 46 | theorem cast_eq_zero [CharZero α] {n : ℚ} : (n : α) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Categ... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | 82 | 86 | theorem piIsoPi_hom_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι)
(x : (∏ᶜ α : TopCat.{max v u})) : (piIsoPi α).hom x i = (Pi.π α i : _) x := by |
have := piIsoPi_inv_π α i
rw [Iso.inv_comp_eq] at this
exact ConcreteCategory.congr_hom this x
|
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.CategoryTheory.Preadditive.Yoneda.Limits
import Mat... | Mathlib/CategoryTheory/Abelian/Injective.lean | 45 | 48 | theorem injective_of_preservesFiniteColimits_preadditiveYonedaObj (J : C)
[hP : PreservesFiniteColimits (preadditiveYonedaObj J)] : Injective J := by |
rw [injective_iff_preservesEpimorphisms_preadditive_yoneda_obj']
infer_instance
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 56 | 103 | theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
(m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m := by |
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩
have neK : ∀ (K : Comp... |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 57 | 58 | theorem star_smul (r : R) (x : CliffordAlgebra Q) : star (r • x) = r • star x := by |
rw [star_def, star_def, map_smul, map_smul]
|
/-
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.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 84 | 86 | theorem measure_iUnion_fintype_le [Fintype ι] (μ : F) (s : ι → Set α) :
μ (⋃ i, s i) ≤ ∑ i, μ (s i) := by |
simpa using measure_biUnion_finset_le Finset.univ s
|
/-
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.ExtendFrom
import Mathlib.Topology.Order.DenselyOrdered
#align_import topology.algebra.order.extend_from fro... | Mathlib/Topology/Order/ExtendFrom.lean | 23 | 33 | theorem continuousOn_Icc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β}
(hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la))
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : Cont... |
apply continuousOn_extendFrom
· rw [closure_Ioo hab]
· intro x x_in
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl | rfl | h)
· exact ⟨la, ha.mono_left <| nhdsWithin_mono _ Ioo_subset_Ioi_self⟩
· exact ⟨lb, hb.mono_left <| nhdsWithin_mono _ Ioo_subset_Iio_self⟩
· exact ⟨f x, hf x h⟩
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.NormedSpace.SphereNormEquiv
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Measure... | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | 62 | 63 | theorem toSphere_apply_univ' : μ.toSphere univ = dim E * μ (ball 0 1 \ {0}) := by |
rw [μ.toSphere_apply' .univ, image_univ, Subtype.range_coe, Ioo_smul_sphere_zero] <;> 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.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib... | Mathlib/Analysis/Complex/Basic.lean | 58 | 59 | theorem norm_exp_ofReal_mul_I (t : ℝ) : ‖exp (t * I)‖ = 1 := by |
simp only [norm_eq_abs, abs_exp_ofReal_mul_I]
|
/-
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 | 144 | 145 | theorem quadraticChar_zero : quadraticChar F 0 = 0 := by |
simp only [quadraticChar_apply, quadraticCharFun_zero]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 83 | 84 | theorem intervalIntegrable_iff : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ι a b) μ := by |
rw [uIoc_eq_union, integrableOn_union, IntervalIntegrable]
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.big_operato... | Mathlib/Algebra/BigOperators/Pi.lean | 89 | 94 | theorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)
(H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by |
cases nonempty_fintype I
ext k
rw [← Finset.univ_prod_mulSingle k, map_prod, map_prod]
simp only [H]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | Mathlib/FieldTheory/RatFunc/Degree.lean | 102 | 107 | theorem natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree {x : RatFunc K}
(hx : x ≠ 0) {s : K[X]} (hs : s ≠ 0) :
((x.num * s).natDegree : ℤ) - (s * x.denom).natDegree = x.intDegree := by |
apply natDegree_sub_eq_of_prod_eq (mul_ne_zero (num_ne_zero hx) hs)
(mul_ne_zero hs x.denom_ne_zero) (num_ne_zero hx) x.denom_ne_zero
rw [mul_assoc]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.RingTheory.Ideal.Maps
#align_import algebra.algebra.subalgebra.basic from "leanprover-c... | Mathlib/Algebra/Algebra/Subalgebra/Operations.lean | 40 | 68 | theorem mem_of_finset_sum_eq_one_of_pow_smul_mem
{ι : Type*} (ι' : Finset ι) (s : ι → S) (l : ι → S)
(e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ i, ∃ n : ℕ, (s i ^ n : S) • x ∈ S') : x ∈ S' := by |
-- Porting note: needed to add this instance
let _i : Algebra { x // x ∈ S' } { x // x ∈ S' } := Algebra.id _
suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range by
obtain ⟨x, rfl⟩ := this
exact x.2
choose n hn using H
let s' : ι → S' := fun x => ⟨s x, hs x⟩
let l' : ι → S' := fun x => ⟨l... |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombinat... | Mathlib/NumberTheory/FLT/Four.lean | 72 | 85 | theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by |
let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
have S_nonempty : S.Nonempty := by
use Int.natAbs c
rw [Set.mem_setOf_eq]
use ⟨a, ⟨b, c⟩⟩
let m : ℕ := Nat.find S_nonempty
have m_mem : m ∈ S := Nat.find_spec S_nonempty
rcases m_mem with ⟨s0, hs0, hs1⟩
us... |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Ba... | Mathlib/RingTheory/AlgebraicIndependent.lean | 134 | 135 | theorem coe_range : AlgebraicIndependent R ((↑) : range x → A) := by |
simpa using hx.comp _ (rangeSplitting_injective x)
|
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.... | Mathlib/Topology/Algebra/Order/Field.lean | 63 | 67 | theorem Filter.Tendsto.atTop_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atTop)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by |
refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC))
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)), hf.eventually_ge_atTop 0]
with x hg hf using mul_le_mul_of_nonneg_left hg.le hf
|
/-
Copyright (c) 2020 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Lacker, Bryan Gin-ge Chen
-/
import Mathlib.Data.Nat.Prime
#align_import data.int.nat_prime from "leanprover-community/mathlib"@"422e70f7ce183d2900c586a8cda8381e788a0c62"
/-... | Mathlib/Data/Int/NatPrime.lean | 36 | 39 | theorem Prime.dvd_natAbs_of_coe_dvd_sq {p : ℕ} (hp : p.Prime) (k : ℤ) (h : (p : ℤ) ∣ k ^ 2) :
p ∣ k.natAbs := by |
apply @Nat.Prime.dvd_of_dvd_pow _ _ 2 hp
rwa [sq, ← natAbs_mul, ← natCast_dvd, ← sq]
|
/-
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.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprove... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 83 | 108 | theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
(hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
(algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by |
rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
one_mul] at hx
replace hx := eq_neg_of_add_eq_zero_left hx
have : ∀ n < f.natDegree, p ∣ f.coeff n := by
intro n hn
exact mem_span_s... |
/-
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.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 112 | 113 | theorem contDiffAt_of_subsingleton [Subsingleton F] : ContDiffAt 𝕜 n f x := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiffAt_const
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Extensions of Groups
This file defines the HNN extension of a group `G`, `HNN... | Mathlib/GroupTheory/HNNExtension.lean | 81 | 83 | theorem inv_t_mul_of (b : B) :
t⁻¹ * (of (b : G) : HNNExtension G A B φ) = of (φ.symm b : G) * t⁻¹ := by |
rw [equiv_symm_eq_conj]; simp
|
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 123 | 132 | theorem xgcdAux_P {r r'} :
∀ {s t s' t'}, P x y (r, s, t) → P x y (r', s', t') → P x y (xgcdAux r s t r' s' t') := by |
induction r, r' using gcd.induction with
| H0 => simp
| H1 a b h IH =>
intro s t s' t' p p'
rw [xgcdAux_rec h]; refine IH ?_ p; dsimp [P] at *
rw [Int.emod_def]; generalize (b / a : ℤ) = k
rw [p, p', Int.mul_sub, sub_add_eq_add_sub, Int.mul_sub, Int.add_mul, mul_comm k t,
mul_comm k s, ← mu... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "le... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 84 | 89 | theorem isCompactOperator_iff_exists_mem_nhds_isCompact_closure_image [T2Space M₂] (f : M₁ → M₂) :
IsCompactOperator f ↔ ∃ V ∈ (𝓝 0 : Filter M₁), IsCompact (closure <| f '' V) := by |
rw [isCompactOperator_iff_exists_mem_nhds_image_subset_compact]
exact
⟨fun ⟨V, hV, K, hK, hKV⟩ => ⟨V, hV, hK.closure_of_subset hKV⟩,
fun ⟨V, hV, hVc⟩ => ⟨V, hV, closure (f '' V), hVc, subset_closure⟩⟩
|
/-
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.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_impor... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 109 | 110 | theorem summable_div_const_iff (h : a ≠ 0) : (Summable fun i ↦ f i / a) ↔ Summable f := by |
simpa only [div_eq_mul_inv] using summable_mul_right_iff (inv_ne_zero h)
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.FunctorCategory
import Mathlib.CategoryTheory.Limits.Shapes.FunctorCategory
impo... | Mathlib/CategoryTheory/Abelian/FunctorCategory.lean | 64 | 76 | theorem coimageImageComparison_app :
coimageImageComparison (α.app X) =
(coimageObjIso α X).inv ≫ (coimageImageComparison α).app X ≫ (imageObjIso α X).hom := by |
ext
dsimp
dsimp [imageObjIso, coimageObjIso, cokernel.map]
simp only [coimage_image_factorisation, PreservesKernel.iso_hom, Category.assoc,
kernel.lift_ι, Category.comp_id, PreservesCokernel.iso_inv,
cokernel.π_desc_assoc, Category.id_comp]
erw [kernelComparison_comp_ι _ ((evaluation C D).obj X),
... |
/-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Coalgebras
In this file we define... | Mathlib/RingTheory/Coalgebra/Basic.lean | 65 | 67 | theorem coassoc_symm_apply (a : A) :
(TensorProduct.assoc R A A A).symm (comul.lTensor A (comul a)) = comul.rTensor A (comul a) := by |
rw [(TensorProduct.assoc R A A A).symm_apply_eq, coassoc_apply a]
|
/-
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 | 116 | 117 | theorem Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by |
rw [← interior_Ico, mem_interior_iff_mem_nhds]
|
/-
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.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# Rees alg... | Mathlib/RingTheory/ReesAlgebra.lean | 113 | 123 | theorem reesAlgebra.fg (hI : I.FG) : (reesAlgebra I).FG := by |
classical
obtain ⟨s, hs⟩ := hI
rw [← adjoin_monomial_eq_reesAlgebra, ← hs]
use s.image (monomial 1)
rw [Finset.coe_image]
change
_ =
Algebra.adjoin R
(Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])
rw [Submodule.map_span, Algebra.adjoin_spa... |
/-
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.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
/-!
# Finite products... | Mathlib/Data/Prod/TProd.lean | 90 | 90 | theorem elim_self (v : TProd α (i :: l)) : v.elim (l.mem_cons_self i) = v.1 := by | simp [TProd.elim]
|
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
/-!
# Binomial rings
In this fi... | Mathlib/RingTheory/Binomial.lean | 107 | 115 | theorem descPochhammer_smeval_eq_ascPochhammer (r : R) (n : ℕ) :
(descPochhammer ℤ n).smeval r = (ascPochhammer ℕ n).smeval (r - n + 1) := by |
induction n with
| zero => simp only [descPochhammer_zero, ascPochhammer_zero, smeval_one, npow_zero]
| succ n ih =>
rw [Nat.cast_succ, sub_add, add_sub_cancel_right, descPochhammer_succ_right, smeval_mul, ih,
ascPochhammer_succ_left, X_mul, smeval_mul_X, smeval_comp, smeval_sub, ← C_eq_natCast,
... |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
i... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 148 | 149 | theorem transReflReparamAux_one : transReflReparamAux 1 = 1 := by |
set_option tactic.skipAssignedInstances false in norm_num [transReflReparamAux]
|
/-
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.List.Basic
/-!
# insertNth
Proves various lemmas about `List.insertNth`.... | Mathlib/Data/List/InsertNth.lean | 116 | 119 | theorem insertNth_length_self (l : List α) (x : α) : insertNth l.length x l = l ++ [x] := by |
induction' l with hd tl IH
· simp
· simpa using IH
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.LinearMap
#align_import linear_algebra.matrix.diagonal from "leanprover-community/ma... | Mathlib/LinearAlgebra/Matrix/Diagonal.lean | 86 | 94 | theorem rank_diagonal [DecidableEq m] [DecidableEq K] (w : m → K) :
LinearMap.rank (toLin' (diagonal w)) = Fintype.card { i // w i ≠ 0 } := by |
have hu : univ ⊆ { i : m | w i = 0 }ᶜ ∪ { i : m | w i = 0 } := by rw [Set.compl_union_self]
have hd : Disjoint { i : m | w i ≠ 0 } { i : m | w i = 0 } := disjoint_compl_left
have B₁ := iSup_range_stdBasis_eq_iInf_ker_proj K (fun _ : m => K) hd hu (Set.toFinite _)
have B₂ := iInfKerProjEquiv K (fun _ ↦ K) hd hu... |
/-
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.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Perm/Sign.lean | 113 | 118 | theorem closure_isSwap [Finite α] : Subgroup.closure { σ : Perm α | IsSwap σ } = ⊤ := by |
cases nonempty_fintype α
refine eq_top_iff.mpr fun x _ => ?_
obtain ⟨h1, h2⟩ := Subtype.mem (truncSwapFactors x).out
rw [← h1]
exact Subgroup.list_prod_mem _ fun y hy => Subgroup.subset_closure (h2 y hy)
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, James Gallicchio
-/
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
/-!
# Pairwise relations on a list
This file provides basic results about `List.... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 106 | 106 | theorem pairwise_pair {a b : α} : Pairwise R [a, b] ↔ R a b := by | 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, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 49 | 53 | theorem rpow_def_of_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) :
x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) := by |
simp only [rpow_def, Complex.cpow_def]; split_ifs <;>
simp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -RCLike.ofReal_mul,
(Complex.ofReal_mul _ _).symm, Complex.exp_ofReal_re, Complex.ofReal_eq_zero]
|
/-
Copyright (c) 2022 Sam van Gool and Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sam van Gool, Jake Levinson
-/
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
i... | Mathlib/Topology/Sheaves/LocallySurjective.lean | 78 | 118 | theorem locally_surjective_iff_surjective_on_stalks (T : ℱ ⟶ 𝒢) :
IsLocallySurjective T ↔ ∀ x : X, Function.Surjective ((stalkFunctor C x).map T) := by |
constructor <;> intro hT
· /- human proof:
Let g ∈ Γₛₜ 𝒢 x be a germ. Represent it on an open set U ⊆ X
as ⟨t, U⟩. By local surjectivity, pass to a smaller open set V
on which there exists s ∈ Γ_ ℱ V mapping to t |_ V.
Then the germ of s maps to g -/
-- Let g ∈ Γₛₜ 𝒢 x be a ge... |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-com... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 61 | 76 | theorem affineIndependent_of_toMatrix_right_inv [Fintype ι] [Finite ι'] [DecidableEq ι']
(p : ι' → P) {A : Matrix ι ι' k} (hA : b.toMatrix p * A = 1) : AffineIndependent k p := by |
cases nonempty_fintype ι'
rw [affineIndependent_iff_eq_of_fintype_affineCombination_eq]
intro w₁ w₂ hw₁ hw₂ hweq
have hweq' : w₁ ᵥ* b.toMatrix p = w₂ ᵥ* b.toMatrix p := by
ext j
change (∑ i, w₁ i • b.coord j (p i)) = ∑ i, w₂ i • b.coord j (p i)
-- Porting note: Added `u` because `∘` was causing tro... |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 63 | 71 | theorem separable_minpoly_mod {p : ℕ} [Fact p.Prime] (hdiv : ¬p ∣ n) :
Separable (map (Int.castRingHom (ZMod p)) (minpoly ℤ μ)) := by |
have hdvd : map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ X ^ n - 1 := by
convert RingHom.map_dvd (mapRingHom (Int.castRingHom (ZMod p)))
(minpoly_dvd_x_pow_sub_one h)
simp only [map_sub, map_pow, coe_mapRingHom, map_X, map_one]
refine Separable.of_dvd (separable_X_pow_sub_C 1 ?_ one_ne_zero) hdvd... |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
/-!
# Binomial rings
In this fi... | Mathlib/RingTheory/Binomial.lean | 101 | 103 | theorem ascPochhammer_smeval_eq_eval [Semiring R] (r : R) (n : ℕ) :
(ascPochhammer ℕ n).smeval r = (ascPochhammer R n).eval r := by |
rw [eval_eq_smeval, ascPochhammer_smeval_cast R]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-com... | Mathlib/Probability/Kernel/WithDensity.lean | 108 | 113 | theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
(hf : Measurable (Function.uncurry f)) (a : α) {g : β → ℝ≥0∞} (hg : Measurable g) :
∫⁻ b, g b ∂withDensity κ f a = ∫⁻ b, f a b * g b ∂κ a := by |
rw [kernel.withDensity_apply _ hf,
lintegral_withDensity_eq_lintegral_mul _ (Measurable.of_uncurry_left hf) hg]
simp_rw [Pi.mul_apply]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.Numb... | Mathlib/NumberTheory/PellMatiyasevic.lean | 151 | 151 | theorem xn_one : xn a1 1 = a := by | simp
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mat... | Mathlib/Analysis/Complex/RealDeriv.lean | 49 | 62 | theorem HasStrictDerivAt.real_of_complex (h : HasStrictDerivAt e e' z) :
HasStrictDerivAt (fun x : ℝ => (e x).re) e'.re z := by |
have A : HasStrictFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasStrictFDerivAt
have B :
HasStrictFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ)
(ofRealCLM z) :=
h.hasStrictFDerivAt.restrictScalars ℝ
have C : HasStrictFDerivAt re reCLM (e (ofRealCLM z)) := reCLM... |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd29... | Mathlib/CategoryTheory/EqToHom.lean | 126 | 129 | theorem congrArg_cast_hom_right {X Y Z : C} (p : X ⟶ Y) (q : Z = Y) :
cast (congrArg (fun W : C => X ⟶ W) q.symm) p = p ≫ eqToHom q.symm := by |
cases q
simp
|
/-
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 | 57 | 73 | theorem coeff_derivative (p : R[X]) (n : ℕ) :
coeff (derivative p) n = coeff p (n + 1) * (n + 1) := by |
rw [derivative_apply]
simp only [coeff_X_pow, coeff_sum, coeff_C_mul]
rw [sum, Finset.sum_eq_single (n + 1)]
· simp only [Nat.add_succ_sub_one, add_zero, mul_one, if_true, eq_self_iff_true]; norm_cast
· intro b
cases b
· intros
rw [Nat.cast_zero, mul_zero, zero_mul]
· intro _ H
rw [Na... |
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f... | Mathlib/Init/Order/LinearOrder.lean | 61 | 65 | theorem le_max_right (a b : α) : b ≤ max a b := by |
-- Porting note: no `min_tac` tactic
if h : a ≤ b
then simp [max_def, if_pos h, le_refl]
else simp [max_def, if_neg h]; exact le_of_not_le h
|
/-
Copyright (c) 2022 Cuma Kökmen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Cuma Kökmen, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 88 | 89 | theorem torusMap_eq_center_iff {c : ℂⁿ} {R : ℝⁿ} {θ : ℝⁿ} : torusMap c R θ = c ↔ R = 0 := by |
simp [funext_iff, torusMap, exp_ne_zero]
|
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | Mathlib/SetTheory/Game/Nim.lean | 59 | 64 | theorem nim_def (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
nim o =
PGame.mk o.out.α o.out.α (fun o₂ => nim (Ordinal.typein (· < ·) o₂)) fun o₂ =>
nim (Ordinal.typein (· < ·) o₂) := by |
rw [nim]; rfl
|
/-
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.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.functor_gamma from "leanprover... | Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 55 | 61 | theorem iff {j : ℕ} {i : Fin (j + 2)} : Isδ₀ (SimplexCategory.δ i) ↔ i = 0 := by |
constructor
· rintro ⟨_, h₂⟩
by_contra h
exact h₂ (Fin.succAbove_ne_zero_zero h)
· rintro rfl
exact ⟨rfl, by dsimp; exact Fin.succ_ne_zero (0 : Fin (j + 1))⟩
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Unbundled semirin... | Mathlib/Deprecated/Ring.lean | 114 | 115 | theorem map_sub (hf : IsRingHom f) : f (x - y) = f x - f y := by |
simp [sub_eq_add_neg, hf.map_add, hf.map_neg]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mat... | Mathlib/Algebra/Lie/Character.lean | 44 | 45 | theorem lieCharacter_apply_lie (χ : LieCharacter R L) (x y : L) : χ ⁅x, y⁆ = 0 := by |
rw [LieHom.map_lie, LieRing.of_associative_ring_bracket, mul_comm, sub_self]
|
/-
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 | 106 | 109 | theorem heq_eqRec_iff_heq {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') {β : Sort _} (y : β) :
HEq y (@Eq.rec α a motive x a' e) ↔ HEq y x := by |
subst e; rfl
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 118 | 120 | theorem associatedPrimes.eq_empty_of_subsingleton [Subsingleton M] : associatedPrimes R M = ∅ := by |
ext; simp only [Set.mem_empty_iff_false, iff_false_iff];
apply not_isAssociatedPrime_of_subsingleton
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Limits.Preserves.Opposites
import Mathlib.Topology.Category.TopCat.Yoneda
import Mathlib.Condensed.Explicit
/-!
# The functor from... | Mathlib/Condensed/TopComparison.lean | 65 | 86 | theorem equalizerCondition_yonedaPresheaf
[∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) G]
(hq : ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], QuotientMap (G.map π)) :
EqualizerCondition (yonedaPresheaf G X) := by |
apply EqualizerCondition.mk
intro Z B π _ _
refine ⟨fun a b h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩
· simp only [yonedaPresheaf, unop_op, Quiver.Hom.unop_op, Set.coe_setOf, MapToEqualizer,
Set.mem_setOf_eq, Subtype.mk.injEq, comp, ContinuousMap.mk.injEq] at h
simp only [yonedaPresheaf, unop_op]
ext x
obtai... |
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_the... | Mathlib/GroupTheory/Perm/Fin.lean | 44 | 45 | theorem Equiv.Perm.decomposeFin_symm_apply_zero {n : ℕ} (p : Fin (n + 1)) (e : Perm (Fin n)) :
Equiv.Perm.decomposeFin.symm (p, e) 0 = p := by | simp [Equiv.Perm.decomposeFin]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Data.TypeVec
import Mathlib.Logic.Equiv.Defs
#align_import control.functor.multivariate from "... | Mathlib/Control/Functor/Multivariate.lean | 122 | 132 | theorem exists_iff_exists_of_mono {P : F α → Prop} {q : F β → Prop}
(f : α ⟹ β) (g : β ⟹ α)
(h₀ : f ⊚ g = TypeVec.id)
(h₁ : ∀ u : F α, P u ↔ q (f <$$> u)) :
(∃ u : F α, P u) ↔ ∃ u : F β, q u := by |
constructor <;> rintro ⟨u, h₂⟩
· refine ⟨f <$$> u, ?_⟩
apply (h₁ u).mp h₂
· refine ⟨g <$$> u, ?_⟩
rw [h₁]
simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]
|
/-
Copyright (c) 2020 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Jeremy Tan
-/
import Mathlib.Analysis.Complex.AbelLimit
import Mathlib.Analysis.SpecialFunctions.Complex.Arctan
#align_import data.real.pi.leibniz from "leanprov... | Mathlib/Data/Real/Pi/Leibniz.lean | 21 | 57 | theorem tendsto_sum_pi_div_four :
Tendsto (fun k => ∑ i ∈ range k, (-1 : ℝ) ^ i / (2 * i + 1)) atTop (𝓝 (π / 4)) := by |
-- The series is alternating with terms of decreasing magnitude, so it converges to some limit
obtain ⟨l, h⟩ :
∃ l, Tendsto (fun n ↦ ∑ i ∈ range n, (-1 : ℝ) ^ i / (2 * i + 1)) atTop (𝓝 l) := by
apply Antitone.tendsto_alternating_series_of_tendsto_zero
· exact antitone_iff_forall_lt.mpr fun _ _ _ ↦ b... |
/-
Copyright (c) 2023 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Geometry.Manifold.Sheaf.Smooth
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-! # Smooth manifolds as locally ringed spaces
This file equ... | Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean | 102 | 107 | theorem smoothSheafCommRing.nonunits_stalk (x : M) :
nonunits ((smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x)
= RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) := by |
ext1 f
rw [mem_nonunits_iff, not_iff_comm, Iff.comm]
apply smoothSheafCommRing.isUnit_stalk_iff
|
/-
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.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-communi... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 101 | 104 | theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩] := by |
rcases le_one_iff.1 hb with (rfl | rfl)
· exact zero_CNF ho
· exact one_CNF ho
|
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_impor... | Mathlib/RingTheory/ChainOfDivisors.lean | 91 | 95 | theorem first_of_chain_isUnit {q : Associates M} {n : ℕ} {c : Fin (n + 1) → Associates M}
(h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) : IsUnit (c 0) := by |
obtain ⟨i, hr⟩ := h₂.mp Associates.one_le
rw [Associates.isUnit_iff_eq_one, ← Associates.le_one_iff, hr]
exact h₁.monotone (Fin.zero_le i)
|
/-
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.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 60 | 62 | theorem χ₄_int_mod_four (n : ℤ) : χ₄ n = χ₄ (n % 4 : ℤ) := by |
rw [← ZMod.intCast_mod n 4]
norm_cast
|
/-
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.Combinatorics.SimpleGraph.Basic
/-!
# Darts in graphs
A `Dart` or half-edge or bond in a graph is an ordered pair of adjacent vertices, regarded as an
orie... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 118 | 123 | theorem dart_edge_eq_mk'_iff' :
∀ {d : G.Dart} {u v : V},
d.edge = s(u, v) ↔ d.fst = u ∧ d.snd = v ∨ d.fst = v ∧ d.snd = u := by |
rintro ⟨⟨a, b⟩, h⟩ u v
rw [dart_edge_eq_mk'_iff]
simp
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Sqrt
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed... | Mathlib/Data/Complex/Abs.lean | 75 | 76 | theorem abs_ofReal (r : ℝ) : Complex.abs r = |r| := by |
simp [Complex.abs, normSq_ofReal, Real.sqrt_mul_self_eq_abs]
|
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 112 | 113 | theorem xgcd_val (x y) : xgcd x y = (gcdA x y, gcdB x y) := by |
unfold gcdA gcdB; cases xgcd x y; rfl
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 43 | 47 | theorem map_mapAccumr (f₁ : β → γ) :
(map f₁ (mapAccumr f₂ xs s).snd) = (mapAccumr (fun x s =>
let r := (f₂ x s); (r.fst, f₁ r.snd)
) xs s).snd := by |
induction xs using Vector.revInductionOn generalizing s <;> simp_all
|
/-
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.Complex.Log
#align_import ana... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 102 | 104 | theorem cpow_neg (x y : ℂ) : x ^ (-y) = (x ^ y)⁻¹ := by |
simp only [cpow_def, neg_eq_zero, mul_neg]
split_ifs <;> simp [exp_neg]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLat... | Mathlib/Algebra/Tropical/BigOperators.lean | 40 | 43 | theorem List.trop_sum [AddMonoid R] (l : List R) : trop l.sum = List.prod (l.map trop) := by |
induction' l with hd tl IH
· simp
· simp [← IH]
|
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
/-!
# Fractional ideal norms
This file defines the absolute ideal norm of a frac... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 90 | 95 | theorem absNorm_eq_zero_iff [NoZeroDivisors K] {I : FractionalIdeal R⁰ K} :
absNorm I = 0 ↔ I = 0 := by |
refine ⟨fun h ↦ zero_of_num_eq_bot zero_not_mem_nonZeroDivisors ?_, fun h ↦ h ▸ absNorm_bot⟩
rw [absNorm_eq, div_eq_zero_iff] at h
refine Ideal.absNorm_eq_zero_iff.mp <| Nat.cast_eq_zero.mp <| h.resolve_right ?_
simpa [Algebra.norm_eq_zero_iff] using nonZeroDivisors.coe_ne_zero _
|
/-
Copyright (c) 2024 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Geoffrey Irving
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.S... | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | 24 | 25 | theorem analyticOn_cexp : AnalyticOn ℂ exp univ := by |
rw [analyticOn_univ_iff_differentiable]; exact differentiable_exp
|
/-
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.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.... | Mathlib/RepresentationTheory/Maschke.lean | 81 | 83 | theorem conjugate_i (g : G) (v : V) : (conjugate π g : W → V) (i v) = v := by |
rw [conjugate_apply, ← i.map_smul, h, ← mul_smul, single_mul_single, mul_one, mul_left_inv,
← one_def, one_smul]
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mat... | Mathlib/NumberTheory/PythagoreanTriples.lean | 37 | 40 | theorem Int.sq_ne_two_mod_four (z : ℤ) : z * z % 4 ≠ 2 := by |
suffices ¬z * z % (4 : ℕ) = 2 % (4 : ℕ) by exact this
rw [← ZMod.intCast_eq_intCast_iff']
simpa using sq_ne_two_fin_zmod_four _
|
/-
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 | 86 | 114 | theorem quasi_compact_affineProperty_iff_quasiSeparatedSpace {X Y : Scheme} [IsAffine Y]
(f : X ⟶ Y) : QuasiCompact.affineProperty.diagonal f ↔ QuasiSeparatedSpace X.carrier := by |
delta AffineTargetMorphismProperty.diagonal
rw [quasiSeparatedSpace_iff_affine]
constructor
· intro H U V
haveI : IsAffine _ := U.2
haveI : IsAffine _ := V.2
let g : pullback (X.ofRestrict U.1.openEmbedding) (X.ofRestrict V.1.openEmbedding) ⟶ X :=
pullback.fst ≫ X.ofRestrict _
-- Porting ... |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprove... | Mathlib/Analysis/Complex/ReImTopology.lean | 109 | 110 | theorem interior_setOf_le_im (a : ℝ) : interior { z : ℂ | a ≤ z.im } = { z | a < z.im } := by |
simpa only [interior_Ici] using interior_preimage_im (Ici a)
|
/-
Copyright (c) 2016 Leonardo de Moura. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor f... | Mathlib/Data/Set/Functor.lean | 149 | 150 | theorem mem_of_mem_image_val (ha : a ∈ (γ : Set α)) : ⟨a, image_val_subset ha⟩ ∈ γ := by |
rcases ha with ⟨_, ha, rfl⟩; exact ha
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 55 | 56 | theorem involute_comp_involute : involute.comp involute = AlgHom.id R (CliffordAlgebra Q) := by |
ext; simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
/-!
# Prepa... | Mathlib/Data/Multiset/FinsetOps.lean | 100 | 117 | theorem attach_ndinsert (a : α) (s : Multiset α) :
(s.ndinsert a).attach =
ndinsert ⟨a, mem_ndinsert_self a s⟩ (s.attach.map fun p => ⟨p.1, mem_ndinsert_of_mem p.2⟩) :=
have eq :
∀ h : ∀ p : { x // x ∈ s }, p.1 ∈ s,
(fun p : { x // x ∈ s } => ⟨p.val, h p⟩ : { x // x ∈ s } → { x // x ∈ s }) = id :=... |
intro t ht
by_cases h : a ∈ s
· rw [ndinsert_of_mem h] at ht
subst ht
rw [eq, map_id, ndinsert_of_mem (mem_attach _ _)]
· rw [ndinsert_of_not_mem h] at ht
subst ht
simp [attach_cons, h]
this _ rfl
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Compl... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 115 | 118 | theorem fst : IsBoundedLinearMap 𝕜 fun x : E × F => x.1 := by |
refine (LinearMap.fst 𝕜 E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_left _ _
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Uni Marx
-/
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
i... | Mathlib/CategoryTheory/Category/RelCat.lean | 65 | 66 | theorem rel_comp_apply₂ {X Y Z : RelCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) (z : Z) :
(f ≫ g) x z ↔ ∃ y, f x y ∧ g y z := by | rfl
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
/-!
# Birkhoff average
In this file we define `birkhoffAverage f g n x` to be
$$
\fr... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 44 | 45 | theorem birkhoffAverage_zero (f : α → α) (g : α → M) (x : α) :
birkhoffAverage R f g 0 x = 0 := by | simp [birkhoffAverage]
|
/-
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.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a... | Mathlib/RingTheory/HahnSeries/Addition.lean | 113 | 119 | theorem embDomain_add (f : Γ ↪o Γ') (x y : HahnSeries Γ R) :
embDomain f (x + y) = embDomain f x + embDomain f y := by |
ext g
by_cases hg : g ∈ Set.range f
· obtain ⟨a, rfl⟩ := hg
simp
· simp [embDomain_notin_range hg]
|
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6... | Mathlib/Algebra/ModEq.lean | 102 | 102 | theorem modEq_zero : a ≡ b [PMOD 0] ↔ a = b := by | simp [ModEq, sub_eq_zero, eq_comm]
|
/-
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.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicit... | Mathlib/Algebra/Polynomial/Div.lean | 61 | 82 | theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
(hq : q ≠ 0) : multiplicity.Finite p q :=
have zn0 : (0 : R) ≠ 1 :=
haveI := Nontrivial.of_polynomial_ne hq
zero_ne_one
⟨natDegree q, fun ⟨r, hr⟩ => by
have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at... | simp [show _ = _ from hmp]
have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne,
hr0, not_false_eq_true]
h... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | Mathlib/Topology/Instances/Sign.lean | 38 | 41 | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a := by |
refine (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr ?_
rw [Filter.EventuallyEq, eventually_nhds_iff]
exact ⟨{ x | x < 0 }, fun x hx => (sign_neg hx).symm, isOpen_gt' 0, h⟩
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 90 | 96 | theorem not_isOfFinOrder_of_injective_pow {x : G} (h : Injective fun n : ℕ => x ^ n) :
¬IsOfFinOrder x := by |
simp_rw [isOfFinOrder_iff_pow_eq_one, not_exists, not_and]
intro n hn_pos hnx
rw [← pow_zero x] at hnx
rw [h hnx] at hn_pos
exact irrefl 0 hn_pos
|
/-
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.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264... | Mathlib/Algebra/MvPolynomial/Rename.lean | 109 | 115 | theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by |
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.quotient_pi from "leanprover-community/mathlib"@"398f60f60... | Mathlib/LinearAlgebra/QuotientPi.lean | 99 | 108 | theorem right_inv : Function.RightInverse (invFun p) (toFun p) := by |
dsimp only [toFun, invFun]
rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]
refine congr_arg _ (pi_ext fun i x => Quotient.inductionOn' x fun x' => funext fun j => ?_)
rw [comp_apply, piQuotientLift_single, Quotient.mk''_eq_mk, mapQ_apply,
quotientPiLift_mk, id_apply]
by_cases hij : i = j <;>... |
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