Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 548 | 549 | theorem uniformity_le_symm : 𝓤 α ≤ @Prod.swap α α <$> 𝓤 α := by |
rw [map_swap_eq_comap_swap]; exact tendsto_swap_uniformity.le_comap
|
/-
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, Kevin Klinge, Andrew Yang
-/
import Mathlib.RingTheory.OreLocalization.OreSet
import Mathlib.Algebra.Group.Submonoid.Operations
#align_import ring_theory.ore_local... | Mathlib/RingTheory/OreLocalization/Basic.lean | 557 | 562 | theorem universalMulHom_unique (φ : R[S⁻¹] →* T) (huniv : ∀ r : R, φ (numeratorHom r) = f r) :
φ = universalMulHom f fS hf := by |
ext x; induction' x using OreLocalization.ind with r s
rw [universalMulHom_apply, ← huniv r, numeratorHom_apply, ← one_mul (φ (r /ₒ s)), ←
Units.val_one, ← mul_left_inv (fS s), Units.val_mul, mul_assoc, ← hf, ← huniv, ← φ.map_mul,
numeratorHom_apply, OreLocalization.mul_cancel]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpac... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 123 | 130 | theorem StrictConvexSpace.of_norm_add_ne_two
(h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E := by |
refine
StrictConvexSpace.of_norm_combo_ne_one fun x y hx hy hne =>
⟨1 / 2, 1 / 2, one_half_pos.le, one_half_pos.le, add_halves _, ?_⟩
rw [← smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, ← div_eq_inv_mul, Ne,
div_eq_one_iff_eq (two_ne_zero' ℝ)]
exact h hx hy hne
|
/-
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.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 199 | 222 | theorem degree_sum_eq_of_disjoint (f : S → R[X]) (s : Finset S)
(h : Set.Pairwise { i | i ∈ s ∧ f i ≠ 0 } (Ne on degree ∘ f)) :
degree (s.sum f) = s.sup fun i => degree (f i) := by |
classical
induction' s using Finset.induction_on with x s hx IH
· simp
· simp only [hx, Finset.sum_insert, not_false_iff, Finset.sup_insert]
specialize IH (h.mono fun _ => by simp (config := { contextual := true }))
rcases lt_trichotomy (degree (f x)) (degree (s.sum f)) with (H | H | H)
· rw [← IH,... |
/-
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.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2d... | Mathlib/Algebra/Lie/Abelian.lean | 228 | 229 | theorem coe_maxTrivLinearMapEquivLieModuleHom (f : maxTrivSubmodule R L (M →ₗ[R] N)) :
(maxTrivLinearMapEquivLieModuleHom f : M → N) = f := by | ext; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Range.lean | 437 | 438 | theorem surjective_rangeRestrict : Surjective f.rangeRestrict := by |
rw [← range_eq_top, range_rangeRestrict]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 1,992 | 1,997 | theorem _root_.AddSubmonoid.mem_iSup_iff_exists_dfinsupp' [AddCommMonoid γ] (S : ι → AddSubmonoid γ)
[∀ (i) (x : S i), Decidable (x ≠ 0)] (x : γ) :
x ∈ iSup S ↔ ∃ f : Π₀ i, S i, (f.sum fun i xi => ↑xi) = x := by |
rw [AddSubmonoid.mem_iSup_iff_exists_dfinsupp]
simp_rw [sumAddHom_apply]
rfl
|
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Scott Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanpr... | Mathlib/Data/Matrix/Basis.lean | 139 | 140 | theorem apply_of_row_ne {i i' : m} (hi : i ≠ i') (j j' : n) (a : α) :
stdBasisMatrix i j a i' j' = 0 := by | simp [hi]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 369 | 399 | theorem σ_comp_σ {n} {i j : Fin (n + 1)} (H : i ≤ j) :
σ (Fin.castSucc i) ≫ σ j = σ j.succ ≫ σ i := by |
ext k : 3
dsimp [σ]
cases' k using Fin.lastCases with k
· simp only [len_mk, Fin.predAbove_right_last]
· cases' k using Fin.cases with k
· rw [Fin.castSucc_zero, Fin.predAbove_of_le_castSucc _ 0 (Fin.zero_le _),
Fin.predAbove_of_le_castSucc _ _ (Fin.zero_le _), Fin.castPred_zero,
Fin.predAbov... |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 1,118 | 1,126 | theorem AddValuation.map_mul (x y : ℚ_[p]) :
addValuationDef (x * y : ℚ_[p]) = addValuationDef x + addValuationDef y := by |
simp only [addValuationDef]
by_cases hx : x = 0
· rw [hx, if_pos (Eq.refl _), zero_mul, if_pos (Eq.refl _), WithTop.top_add]
· by_cases hy : y = 0
· rw [hy, if_pos (Eq.refl _), mul_zero, if_pos (Eq.refl _), WithTop.add_top]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← WithTop.coe_add, With... |
/-
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.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathl... | Mathlib/Order/JordanHolder.lean | 393 | 394 | theorem length_eq {s₁ s₂ : CompositionSeries X} (h : Equivalent s₁ s₂) : s₁.length = s₂.length := by |
simpa using Fintype.card_congr h.choose
|
/-
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.Algebra.Module.Zlattice.Basic
/-!
# Covolume of ℤ-lattices
Let `E` be a finite dimensional real vector space with an inner product.
Let `L` be a `ℤ`-l... | Mathlib/Algebra/Module/Zlattice/Covolume.lean | 67 | 76 | theorem covolume_eq_det_mul_measure {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ L)
(b₀ : Basis ι ℝ E) :
covolume L μ = |b₀.det ((↑) ∘ b)| * (μ (Zspan.fundamentalDomain b₀)).toReal := by |
rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain b μ),
Zspan.measure_fundamentalDomain _ _ b₀, measure_congr
(Zspan.fundamentalDomain_ae_parallelepiped b₀ μ), ENNReal.toReal_mul, ENNReal.toReal_ofReal
(by positivity)]
congr
ext
exact b.ofZlatticeBasis_apply ℝ L _
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 239 | 240 | theorem front_eq (s : String) : front s = s.1.headD default := by |
unfold front; exact get_of_valid [] s.1
|
/-
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.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Order.Ring.Finset
import Mat... | Mathlib/Analysis/Normed/Field/Basic.lean | 983 | 988 | theorem punctured_nhds_neBot (x : α) : NeBot (𝓝[≠] x) := by |
rw [← mem_closure_iff_nhdsWithin_neBot, Metric.mem_closure_iff]
rintro ε ε0
rcases exists_norm_lt α ε0 with ⟨b, hb0, hbε⟩
refine ⟨x + b, mt (Set.mem_singleton_iff.trans add_right_eq_self).1 <| norm_pos_iff.1 hb0, ?_⟩
rwa [dist_comm, dist_eq_norm, add_sub_cancel_left]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 120 | 121 | theorem smul_of [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R) :
a • of f x = of f (a • x) := by | rw [of, RingHom.comp_apply, RingHom.comp_apply, smul_mk, smul_C]
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
/--
A `Heap` is the nodes of the pairing heap.
Each node have two pointers: `child` going to the first c... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 107 | 111 | theorem Heap.noSibling_tail? {s : Heap α} : s.tail? le = some s' →
s'.NoSibling := by |
simp only [Heap.tail?]; intro eq
match eq₂ : s.deleteMin le, eq with
| some (a, tl), rfl => exact noSibling_deleteMin eq₂
|
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 2,295 | 2,295 | theorem ite_right (s t : Set α) : s.ite t s = t ∩ s := by | simp [Set.ite]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
/-!... | Mathlib/Order/Interval/Multiset.lean | 287 | 290 | theorem Ico_cons_right (h : a ≤ b) : b ::ₘ Ico a b = Icc a b := by |
classical
rw [Ico, ← Finset.insert_val_of_not_mem right_not_mem_Ico, Finset.Ico_insert_right h]
rfl
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e... | Mathlib/Data/Set/Opposite.lean | 92 | 96 | theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by |
ext
constructor
· apply op_injective
· apply unop_injective
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mat... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 70 | 70 | theorem map_comp (g : β → γ) : (p.map f).map g = p.map (g ∘ f) := by | simp [map, Function.comp]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | Mathlib/Algebra/Polynomial/Laurent.lean | 431 | 437 | theorem reduce_to_polynomial_of_mul_T (f : R[T;T⁻¹]) {Q : R[T;T⁻¹] → Prop}
(Qf : ∀ f : R[X], Q (toLaurent f)) (QT : ∀ f, Q (f * T 1) → Q f) : Q f := by |
induction' f using LaurentPolynomial.induction_on_mul_T with f n
induction' n with n hn
· simpa only [Nat.zero_eq, Nat.cast_zero, neg_zero, T_zero, mul_one] using Qf _
· convert QT _ _
simpa using hn
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
/-!
# N-ary maps of fil... | Mathlib/Order/Filter/NAry.lean | 223 | 227 | theorem map₂_right_comm {m : δ → γ → ε} {n : α → β → δ} {m' : α → γ → δ'} {n' : δ' → β → ε}
(h_right_comm : ∀ a b c, m (n a b) c = n' (m' a c) b) :
map₂ m (map₂ n f g) h = map₂ n' (map₂ m' f h) g := by |
rw [map₂_swap n, map₂_swap n']
exact map₂_assoc fun _ _ _ => h_right_comm _ _ _
|
/-
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.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 500 | 502 | theorem comap_exp_nhdsWithin_zero : comap exp (𝓝[≠] 0) = comap re atBot := by |
have : (exp ⁻¹' {0})ᶜ = Set.univ := eq_univ_of_forall exp_ne_zero
simp [nhdsWithin, comap_exp_nhds_zero, this]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 614 | 618 | theorem mem_iInf_of_iInter {ι} {s : ι → Filter α} {U : Set α} {I : Set ι} (I_fin : I.Finite)
{V : I → Set α} (hV : ∀ i, V i ∈ s i) (hU : ⋂ i, V i ⊆ U) : U ∈ ⨅ i, s i := by |
haveI := I_fin.fintype
refine mem_of_superset (iInter_mem.2 fun i => ?_) hU
exact mem_iInf_of_mem (i : ι) (hV _)
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 634 | 635 | theorem rightUnitor_tensor (X Y : C) :
(ρ_ (X ⊗ Y)).hom = (α_ X Y (𝟙_ C)).hom ≫ X ◁ (ρ_ Y).hom := by | simp
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 182 | 184 | theorem add_tsub_add_le_tsub_left : a + b - (a + c) ≤ b - c := by |
rw [tsub_le_iff_left, add_assoc]
exact add_le_add_left le_add_tsub _
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 285 | 292 | theorem rotation_eq_self_iff_angle_eq_zero {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
o.rotation θ x = x ↔ θ = 0 := by |
constructor
· intro h
rw [eq_comm]
simpa [hx, h] using o.oangle_rotation_right hx hx θ
· intro h
simp [h]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,059 | 1,067 | theorem norm_sq_eq_add_norm_sq_projection (x : E) (S : Submodule 𝕜 E) [HasOrthogonalProjection S] :
‖x‖ ^ 2 = ‖orthogonalProjection S x‖ ^ 2 + ‖orthogonalProjection Sᗮ x‖ ^ 2 :=
calc
‖x‖ ^ 2 = ‖(orthogonalProjection S x : E) + orthogonalProjection Sᗮ x‖ ^ 2 := by |
rw [orthogonalProjection_add_orthogonalProjection_orthogonal]
_ = ‖orthogonalProjection S x‖ ^ 2 + ‖orthogonalProjection Sᗮ x‖ ^ 2 := by
simp only [sq]
exact norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero _ _ <|
(S.mem_orthogonal _).1 (orthogonalProjection Sᗮ x).2 _ (orthogonalProjec... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 359 | 376 | theorem setToSimpleFunc_congr' (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
(hf : Integrable f μ) (hg : Integrable g μ)
(h : Pairwise fun x y => T (f ⁻¹' {x} ∩ g ⁻¹' {y}) = 0) :
f.setToSimpleFunc T = g.setToSimpleFunc T :=
show ((pair f g).map Prod.fst).setToSimpleFunc T = ((pair f... |
have h_eq : T ((⇑(f.pair g)) ⁻¹' {(f a, g a)}) = T (f ⁻¹' {f a} ∩ g ⁻¹' {g a}) := by
congr; rw [pair_preimage_singleton f g]
rw [h_eq]
exact h eq
simp only [this, ContinuousLinearMap.zero_apply, pair_apply]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 391 | 399 | theorem setToSimpleFunc_congr_left (T T' : Set α → E →L[ℝ] F)
(h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →ₛ E) (hf : Integrable f μ) :
setToSimpleFunc T f = setToSimpleFunc T' f := by |
simp_rw [setToSimpleFunc]
refine sum_congr rfl fun x _ => ?_
by_cases hx0 : x = 0
· simp [hx0]
· rw [h (f ⁻¹' {x}) (SimpleFunc.measurableSet_fiber _ _)
(SimpleFunc.measure_preimage_lt_top_of_integrable _ hf hx0)]
|
/-
Copyright (c) 2020 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.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 291 | 297 | theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
(hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by |
simp only [tangentMapWithin]
rw [mfderivWithin_fst]
· rcases p with ⟨⟩; rfl
· exact hs
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 557 | 558 | theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by |
simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Bas... | Mathlib/CategoryTheory/Simple.lean | 193 | 201 | theorem Biprod.isIso_inl_iff_isZero (X Y : C) : IsIso (biprod.inl : X ⟶ X ⊞ Y) ↔ IsZero Y := by |
rw [biprod.isIso_inl_iff_id_eq_fst_comp_inl, ← biprod.total, add_right_eq_self]
constructor
· intro h
replace h := h =≫ biprod.snd
simpa [← IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y)] using h
· intro h
rw [IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y)] at h
rw [h, zero_comp... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 1,077 | 1,080 | theorem div_le_div_flip {α : Type*} [CommGroup α] [LinearOrder α]
[CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} : a / b ≤ b / a ↔ a ≤ b := by |
rw [div_eq_mul_inv b, mul_comm]
exact div_le_inv_mul_iff
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 1,441 | 1,453 | theorem pushout.congrHom_inv {X Y Z : C} {f₁ f₂ : X ⟶ Y} {g₁ g₂ : X ⟶ Z} (h₁ : f₁ = f₂)
(h₂ : g₁ = g₂) [HasPushout f₁ g₁] [HasPushout f₂ g₂] :
(pushout.congrHom h₁ h₂).inv =
pushout.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) (by simp [h₁]) (by simp [h₂]) := by |
ext
· erw [pushout.inl_desc]
rw [Iso.comp_inv_eq, Category.id_comp]
erw [pushout.inl_desc]
rw [Category.id_comp]
· erw [pushout.inr_desc]
rw [Iso.comp_inv_eq, Category.id_comp]
erw [pushout.inr_desc]
rw [Category.id_comp]
|
/-
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.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 767 | 771 | theorem mem_insert_self {t : RBNode α} (ht : Balanced t c n) : v ∈ t.insert cmp v := by |
rw [← mem_toList, List.mem_iff_append]
exact match e : zoom (cmp v) t with
| (nil, p) => let ⟨_, _, _, h⟩ := exists_insert_toList_zoom_nil ht e; ⟨_, _, h⟩
| (node .., p) => let ⟨_, _, _, h⟩ := exists_insert_toList_zoom_node ht e; ⟨_, _, h⟩
|
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset... | Mathlib/GroupTheory/Exponent.lean | 682 | 690 | theorem Commute.of_orderOf_dvd_two [IsCancelMul G] (h : ∀ g : G, orderOf g ∣ 2) (a b : G) :
Commute a b := by |
simp_rw [orderOf_dvd_iff_pow_eq_one] at h
rw [commute_iff_eq, ← mul_right_inj a, ← mul_left_inj b]
calc
a * (a * b) * b = a ^ 2 * b ^ 2 := by simp only [pow_two]; group
_ = 1 := by rw [h, h, mul_one]
_ = (a * b) ^ 2 := by rw [h]
_ = a * (b * a) * b := by simp only [pow_two]; group
|
/-
Copyright (c) 2019 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.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
import Mathlib.RingTheory.Adjoin.... | Mathlib/RingTheory/IntegralClosure.lean | 342 | 349 | theorem RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
(hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z := by |
letI : Algebra R S := f.toAlgebra
have := (IsIntegral.fg_adjoin_singleton hx).mul (IsIntegral.fg_adjoin_singleton hy)
rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
exact
IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
(Algebra.mem_adjoin_iff.2 <| Subring.closure_mono... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 640 | 650 | theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
Inducing (comap (algebraMap R S)) := by |
refine ⟨TopologicalSpace.ext_isClosed fun Z ↦ ?_⟩
simp_rw [isClosed_induced_iff, isClosed_iff_zeroLocus, @eq_comm _ _ (zeroLocus _),
exists_exists_eq_and, preimage_comap_zeroLocus]
constructor
· rintro ⟨s, rfl⟩
refine ⟨(Ideal.span s).comap (algebraMap R S), ?_⟩
rw [← zeroLocus_span, ← zeroLocus_spa... |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 173 | 173 | theorem coe_pi_add_coe_pi : (π : Real.Angle) + π = 0 := by | rw [← two_nsmul, two_nsmul_coe_pi]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.Measur... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 491 | 507 | theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
(hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 := by |
let t := hf.sigmaFiniteSet
suffices f =ᵐ[μ.restrict t] 0 from
ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl
haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
refine ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite ?_ ?_
· intro s hs hμs
rw [IntegrableOn, Measure.rest... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,378 | 1,381 | theorem extChartAt_image_nhd_mem_nhds_of_boundaryless [I.Boundaryless]
{x : M} (hx : s ∈ 𝓝 x) : extChartAt I x '' s ∈ 𝓝 (extChartAt I x x) := by |
rw [extChartAt]
exact extend_image_nhd_mem_nhds_of_boundaryless _ I (mem_chart_source H x) hx
|
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | Mathlib/Algebra/Ring/Defs.lean | 352 | 352 | theorem mul_neg_one (a : α) : a * -1 = -a := by | simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"... | Mathlib/Data/Sum/Basic.lean | 332 | 340 | theorem elim_update_left [DecidableEq α] [DecidableEq β] (f : α → γ) (g : β → γ) (i : α) (c : γ) :
Sum.elim (Function.update f i c) g = Function.update (Sum.elim f g) (inl i) c := by |
ext x
rcases x with x | x
· by_cases h : x = i
· subst h
simp
· simp [h]
· simp
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin, Scott Morrison
-/
import Mathlib.Analysis.Normed.Group.SemiNormedGroupCat
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.CategoryTheory.L... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean | 412 | 419 | theorem ExplicitCoker.map_desc {A B C D B' D' : SemiNormedGroupCat.{u}}
{fab : A ⟶ B} {fbd : B ⟶ D} {fac : A ⟶ C} {fcd : C ⟶ D} {h : fab ≫ fbd = fac ≫ fcd}
{fbb' : B ⟶ B'} {fdd' : D ⟶ D'} {condb : fab ≫ fbb' = 0} {condd : fcd ≫ fdd' = 0} {g : B' ⟶ D'}
(h' : fbb' ≫ g = fbd ≫ fdd') :
explicitCokernelDesc ... |
delta explicitCokernel.map
simp [← cancel_epi (explicitCokernelπ fab), ← Category.assoc, Category.assoc,
explicitCokernelπ_desc, h']
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.ZMod.Basic
#align_import data.zmod.parity from "leanprover-community/mathlib"@"048240e809f04e2bde02482ab44bc23074... | Mathlib/Data/ZMod/Parity.lean | 28 | 29 | theorem eq_one_iff_odd {n : ℕ} : (n : ZMod 2) = 1 ↔ Odd n := by |
rw [← @Nat.cast_one (ZMod 2), ZMod.eq_iff_modEq_nat, Nat.odd_iff, Nat.ModEq]
|
/-
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.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
/-!
# Derivatives of aff... | Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | 32 | 34 | theorem hasStrictDerivAt : HasStrictDerivAt f (f.linear 1) x := by |
rw [f.decomp]
exact f.linear.hasStrictDerivAt.add_const (f 0)
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 330 | 332 | theorem get_map (f : α → β) {o : Option α} (h : isSome (o.map f)) :
(o.map f).get h = f (o.get (by rwa [← isSome_map])) := by |
cases o <;> [simp at h; rfl]
|
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 966 | 967 | theorem eq_iff_abs_toReal_eq_of_sign_eq {θ ψ : Angle} (h : θ.sign = ψ.sign) :
θ = ψ ↔ |θ.toReal| = |ψ.toReal| := by | simpa [h] using @eq_iff_sign_eq_and_abs_toReal_eq θ ψ
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 53 | 53 | theorem logb_one : logb b 1 = 0 := by | simp [logb]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 522 | 525 | theorem shortestPrefixDiff_pos {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty)
{x : ∀ n, E n} (hx : x ∉ s) : 0 < shortestPrefixDiff x s := by |
rcases hne with ⟨y, hy⟩
exact (zero_le _).trans_lt (firstDiff_lt_shortestPrefixDiff hs hx hy)
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411... | Mathlib/ModelTheory/ElementaryMaps.lean | 103 | 104 | theorem map_sentence (f : M ↪ₑ[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ := by |
rw [Sentence.Realize, Sentence.Realize, ← f.map_formula, Unique.eq_default (f ∘ default)]
|
/-
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.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdB... | Mathlib/RingTheory/Finiteness.lean | 389 | 415 | theorem fg_iff_compact (s : Submodule R M) : s.FG ↔ CompleteLattice.IsCompactElement s := by |
classical
-- Introduce shorthand for span of an element
let sp : M → Submodule R M := fun a => span R {a}
-- Trivial rewrite lemma; a small hack since simp (only) & rw can't accomplish this smoothly.
have supr_rw : ∀ t : Finset M, ⨆ x ∈ t, sp x = ⨆ x ∈ (↑t : Set M), sp x := fun t => by rfl
constr... |
/-
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.Finset.Lattice
#align_import order.irreducible from "leanprover-community/mathlib"@"bf2428c9486c407ca38b5b3fb10b87dad0bc99fa"
/-!
# Irreducible and ... | Mathlib/Order/Irreducible.lean | 107 | 107 | theorem SupPrime.ne_bot (ha : SupPrime a) : a ≠ ⊥ := by | rintro rfl; exact not_supPrime_bot ha
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 308 | 309 | theorem sqrt_eq_zero' : √x = 0 ↔ x ≤ 0 := by |
rw [sqrt, NNReal.coe_eq_zero, NNReal.sqrt_eq_zero, Real.toNNReal_eq_zero]
|
/-
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.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 1,610 | 1,613 | theorem _root_.Disjoint.of_preimage (hf : Surjective f) {s t : Set β}
(h : Disjoint (f ⁻¹' s) (f ⁻¹' t)) : Disjoint s t := by |
rw [disjoint_iff_inter_eq_empty, ← image_preimage_eq (_ ∩ _) hf, preimage_inter, h.inter_eq,
image_empty]
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 636 | 637 | theorem coe_iInf₂ (f : ∀ i, κ i → UpperSet α) :
(↑(⨅ (i) (j), f i j) : Set α) = ⋃ (i) (j), f i j := by | simp_rw [coe_iInf]
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 440 | 442 | theorem biprod.map_lift_mapBiprod (f : W ⟶ X) (g : W ⟶ Y) :
F.map (biprod.lift f g) ≫ (F.mapBiprod X Y).hom = biprod.lift (F.map f) (F.map g) := by |
ext <;> simp [mapBiprod, ← F.map_comp]
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
/-!
# Non-unital Star Subalgebras
In this... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 710 | 711 | theorem top_toNonUnitalSubalgebra :
(⊤ : NonUnitalStarSubalgebra R A).toNonUnitalSubalgebra = ⊤ := by | ext; simp
|
/-
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.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.fil... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 420 | 430 | theorem biUnionTagged_memBaseSet {π : Prepartition I} {πi : ∀ J, TaggedPrepartition J}
(h : ∀ J ∈ π, l.MemBaseSet J c r (πi J)) (hp : ∀ J ∈ π, (πi J).IsPartition)
(hc : l.bDistortion → π.compl.distortion ≤ c) : l.MemBaseSet I c r (π.biUnionTagged πi) := by |
refine ⟨TaggedPrepartition.isSubordinate_biUnionTagged.2 fun J hJ => (h J hJ).1,
fun hH => TaggedPrepartition.isHenstock_biUnionTagged.2 fun J hJ => (h J hJ).2 hH,
fun hD => ?_, fun hD => ?_⟩
· rw [Prepartition.distortion_biUnionTagged, Finset.sup_le_iff]
exact fun J hJ => (h J hJ).3 hD
· refine ⟨_, ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 415 | 418 | theorem continuousInv_iInf {ts' : ι' → TopologicalSpace G}
(h' : ∀ i, @ContinuousInv G (ts' i) _) : @ContinuousInv G (⨅ i, ts' i) _ := by |
rw [← sInf_range]
exact continuousInv_sInf (Set.forall_mem_range.mpr h')
|
/-
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.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprove... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 262 | 265 | theorem cofan_inj_comp_app (f : X ⟶ Y) {Δ : SimplexCategoryᵒᵖ} (A : IndexSet Δ) :
(s.cofan Δ).inj A ≫ f.app Δ = s.φ f A.1.unop.len ≫ Y.map A.e.op := by |
simp only [cofan_inj_eq_assoc, φ, assoc]
erw [NatTrans.naturality]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 790 | 792 | theorem Ico_union_Ico_eq_Ico {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) :
Ico a b ∪ Ico b c = Ico a c := by |
rw [← coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico_eq_Ico hab hbc]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-communi... | Mathlib/MeasureTheory/Group/Prod.lean | 459 | 464 | theorem quasiMeasurePreserving_div_left_of_right_invariant [IsMulRightInvariant μ] (g : G) :
QuasiMeasurePreserving (fun h : G => g / h) μ μ := by |
rw [← μ.inv_inv]
exact
(quasiMeasurePreserving_div_left μ.inv g).mono (inv_absolutelyContinuous μ.inv)
(absolutelyContinuous_inv μ.inv)
|
/-
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.Data.Nat.ModEq
import Mathlib.Tactic.Abel
import Mathlib.Tactic.GCongr.Core
#align_import data.int.modeq from "leanprover-community/mathlib"@"47a1a73351de... | Mathlib/Data/Int/ModEq.lean | 98 | 100 | theorem modEq_iff_add_fac {a b n : ℤ} : a ≡ b [ZMOD n] ↔ ∃ t, b = a + n * t := by |
rw [modEq_iff_dvd]
exact exists_congr fun t => sub_eq_iff_eq_add'
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 111 | 112 | theorem finSuccEquiv'_at (i : Fin (n + 1)) : (finSuccEquiv' i) i = none := by |
simp [finSuccEquiv']
|
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 142 | 151 | theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by |
ext n
cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, hc]
| succ n =>
have equ : coeff R n (d⁄dX R f) = coeff R n (d⁄dX R g) := by rw [hD]
rwa [coeff_derivative, coeff_derivative, ← cast_succ, mul_comm, ← nsmul_eq_mul,
mul_comm, ← nsmul_eq_mul, smul_right_inj n.succ_ne_zero] at equ
|
/-
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 | 642 | 643 | theorem bihimp_bihimp_bihimp_comm : a ⇔ b ⇔ (c ⇔ d) = a ⇔ c ⇔ (b ⇔ d) := by |
simp_rw [bihimp_assoc, bihimp_left_comm]
|
/-
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 | 168 | 168 | theorem lt_log_iff_exp_lt (hy : 0 < y) : x < log y ↔ exp x < y := by | rw [← exp_lt_exp, exp_log hy]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,465 | 1,510 | theorem integral_withDensity_eq_integral_smul {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E) :
∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ := by |
by_cases hE : CompleteSpace E; swap; · simp [integral, hE]
by_cases hg : Integrable g (μ.withDensity fun x => f x); swap
· rw [integral_undef hg, integral_undef]
rwa [← integrable_withDensity_iff_integrable_smul f_meas]
refine Integrable.induction
(P := fun g => ∫ x, g x ∂μ.withDensity (fun x => f x) =... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 473 | 475 | theorem measure_prod_null {s : Set (α × β)} (hs : MeasurableSet s) :
μ.prod ν s = 0 ↔ (fun x => ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0 := by |
rw [prod_apply hs, lintegral_eq_zero_iff (measurable_measure_prod_mk_left hs)]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 148 | 150 | theorem le_mul_self : a ≤ b * a := by |
rw [mul_comm]
exact le_self_mul
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2f... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 152 | 152 | theorem at_zero : legendreSym p 0 = 0 := by | rw [legendreSym, Int.cast_zero, MulChar.map_zero]
|
/-
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.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 226 | 235 | theorem coeff_mul_monomial (a : R) :
coeff R m (φ * monomial R n a) = if n ≤ m then coeff R (m - n) φ * a else 0 := by |
classical
have :
∀ p ∈ antidiagonal m,
coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 φ * coeff R p.2 (monomial R n a) ≠ 0 → p.2 = n :=
fun p _ hp => eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hp)
rw [coeff_mul, ← Finset.sum_filter_of_ne this, Finset.filter_snd_eq_antidiagonal _ n,
Finset.sum_ite_... |
/-
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.Logic.Function.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.subtype from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185... | Mathlib/Data/Subtype.lean | 183 | 188 | theorem surjective_restrict {α} {β : α → Type*} [ne : ∀ a, Nonempty (β a)] (p : α → Prop) :
Surjective fun f : ∀ x, β x ↦ restrict p f := by |
letI := Classical.decPred p
refine fun f ↦ ⟨fun x ↦ if h : p x then f ⟨x, h⟩ else Nonempty.some (ne x), funext <| ?_⟩
rintro ⟨x, hx⟩
exact dif_pos hx
|
/-
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 | 308 | 311 | theorem Multipliable.tendsto_atTop_one {f : ℕ → G} (hf : Multipliable f) :
Tendsto f atTop (𝓝 1) := by |
rw [← Nat.cofinite_eq_atTop]
exact hf.tendsto_cofinite_one
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 758 | 759 | theorem disjoint_compl_left_iff : Disjoint xᶜ y ↔ y ≤ x := by |
rw [← le_compl_iff_disjoint_left, compl_compl]
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 718 | 723 | theorem ofEquiv_aequiv {U : Type*} [CommRing U] [Algebra R U] (h : IsAdjoinRoot S f)
(h' : IsAdjoinRoot U f) (e : S ≃ₐ[R] T) :
(h.ofEquiv e).aequiv h' = e.symm.trans (h.aequiv h') := by |
ext a
rw [← (h.ofEquiv e).map_repr a, aequiv_map, AlgEquiv.trans_apply, ofEquiv_map_apply,
e.symm_apply_apply, aequiv_map]
|
/-
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.Fintype.Card
import Mathlib.Data.Finset.Option
#align_import data.fintype.option from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11... | Mathlib/Data/Fintype/Option.lean | 114 | 119 | theorem Finite.induction_empty_option {P : Type u → Prop} (of_equiv : ∀ {α β}, α ≃ β → P α → P β)
(h_empty : P PEmpty) (h_option : ∀ {α} [Fintype α], P α → P (Option α)) (α : Type u)
[Finite α] : P α := by |
cases nonempty_fintype α
refine Fintype.induction_empty_option ?_ ?_ ?_ α
exacts [fun α β _ => of_equiv, h_empty, @h_option]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.Action
import Mathlib.Combinatorics.Quiver.Arborescence
import Mathlib.Combinatorics.Quiver.ConnectedComponent
import Mathlib.GroupTheory.FreeGr... | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | 195 | 202 | theorem loopOfHom_eq_id {a b : Generators G} (e) (H : e ∈ wideSubquiverSymmetrify T a b) :
loopOfHom T (of e) = 𝟙 (root' T) := by |
rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]
cases' H with H H
· rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]
rfl
· rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]
simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.complex.circle from "lea... | Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 37 | 38 | theorem arg_expMapCircle {x : ℝ} (h₁ : -π < x) (h₂ : x ≤ π) : arg (expMapCircle x) = x := by |
rw [expMapCircle_apply, exp_mul_I, arg_cos_add_sin_mul_I ⟨h₁, h₂⟩]
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | Mathlib/Data/Int/CardIntervalMod.lean | 112 | 123 | theorem count_modEq_card_eq_ceil (v : ℕ) :
b.count (· ≡ v [MOD r]) = ⌈(b - (v % r : ℕ)) / (r : ℚ)⌉ := by |
have hr' : 0 < (r : ℚ) := by positivity
rw [count_eq_card_filter_range, ← Ico_zero_eq_range, Ico_filter_modEq_card _ _ hr,
max_eq_left (sub_nonneg.mpr <| by gcongr <;> positivity)]
conv_lhs =>
rw [← div_add_mod v r, cast_add, cast_mul, add_comm]
tactic => simp_rw [← sub_sub, sub_div (_ - _), mul_div_... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 574 | 592 | theorem summable_of_ratio_norm_eventually_le {α : Type*} [SeminormedAddCommGroup α]
[CompleteSpace α] {f : ℕ → α} {r : ℝ} (hr₁ : r < 1)
(h : ∀ᶠ n in atTop, ‖f (n + 1)‖ ≤ r * ‖f n‖) : Summable f := by |
by_cases hr₀ : 0 ≤ r
· rw [eventually_atTop] at h
rcases h with ⟨N, hN⟩
rw [← @summable_nat_add_iff α _ _ _ _ N]
refine .of_norm_bounded (fun n ↦ ‖f N‖ * r ^ n)
(Summable.mul_left _ <| summable_geometric_of_lt_one hr₀ hr₁) fun n ↦ ?_
simp only
conv_rhs => rw [mul_comm, ← zero_add N]
r... |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 535 | 536 | theorem pullback_inter {f : Y ⟶ X} (S R : Sieve X) :
(S ⊓ R).pullback f = S.pullback f ⊓ R.pullback f := by | simp [Sieve.ext_iff]
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversa... | Mathlib/Control/Traversable/Instances.lean | 47 | 51 | theorem Option.naturality {α β} (f : α → F β) (x : Option α) :
η (Option.traverse f x) = Option.traverse (@η _ ∘ f) x := by |
-- Porting note: added `ApplicativeTransformation` theorems
cases' x with x <;> simp! [*, functor_norm, ApplicativeTransformation.preserves_map,
ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]
|
/-
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.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization... | Mathlib/RingTheory/LocalProperties.lean | 290 | 300 | theorem eq_zero_of_localization (r : R)
(h : ∀ (J : Ideal R) (hJ : J.IsMaximal), algebraMap R (Localization.AtPrime J) r = 0) :
r = 0 := by |
rw [← Ideal.span_singleton_eq_bot]
apply ideal_eq_bot_of_localization
intro J hJ
delta IsLocalization.coeSubmodule
erw [Submodule.map_span, Submodule.span_eq_bot]
rintro _ ⟨_, h', rfl⟩
cases Set.mem_singleton_iff.mpr h'
exact h J hJ
|
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.co... | Mathlib/Order/CompleteLatticeIntervals.lean | 162 | 168 | theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by |
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine hs.out B.2 c.2 ⟨?_, ?_⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
|
/-
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.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import... | Mathlib/RingTheory/Filtration.lean | 216 | 223 | theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by |
obtain ⟨n₀, hn⟩ := h
use n₀
intro k
induction' k with _ ih
· simp
· rw [← add_assoc, ← hn, ih, add_comm, pow_add, mul_smul, pow_one]
omega
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 581 | 582 | theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b := by |
rw [← coe_inj, coe_insert, coe_Ioc, coe_Icc, Set.insert_eq, Set.union_comm, Set.Ioc_union_left h]
|
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.D... | Mathlib/CategoryTheory/CofilteredSystem.lean | 319 | 323 | theorem toEventualRanges_nonempty (h : F.IsMittagLeffler) [∀ j : J, Nonempty (F.obj j)] (j : J) :
Nonempty (F.toEventualRanges.obj j) := by |
let ⟨i, f, h⟩ := F.isMittagLeffler_iff_eventualRange.1 h j
rw [toEventualRanges_obj, h]
infer_instance
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 203 | 209 | theorem sum_coord_apply_eq_one [Fintype ι] (q : P) : ∑ i, b.coord i q = 1 := by |
have hq : q ∈ affineSpan k (range b) := by
rw [b.tot]
exact AffineSubspace.mem_top k V q
obtain ⟨w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan_of_fintype hq
convert hw
exact b.coord_apply_combination_of_mem (Finset.mem_univ _) hw
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 270 | 281 | theorem starAlgHomClass_ext [T2Space B] {F : Type*} {a : A}
[FunLike F (elementalStarAlgebra R a) B] [AlgHomClass F R _ B] [StarAlgHomClass F R _ B]
{φ ψ : F} (hφ : Continuous φ)
(hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ := by |
-- Note: help with unfolding `elementalStarAlgebra`
have : StarAlgHomClass F R (↥(topologicalClosure (adjoin R {a}))) B :=
inferInstanceAs (StarAlgHomClass F R (elementalStarAlgebra R a) B)
refine StarAlgHomClass.ext_topologicalClosure hφ hψ fun x => ?_
refine adjoin_induction' x ?_ ?_ ?_ ?_ ?_
exacts [f... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstance... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 122 | 125 | theorem subsingleton_iff_eq_bot : Subsingleton p ↔ p = ⊥ := by |
rw [subsingleton_iff, Submodule.eq_bot_iff]
refine ⟨fun h x hx ↦ by simpa using h ⟨x, hx⟩ ⟨0, p.zero_mem⟩,
fun h ⟨x, hx⟩ ⟨y, hy⟩ ↦ by simp [h x hx, h y hy]⟩
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Mario Carneiro, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Algebra.Module.MinimalAxioms
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.... | Mathlib/Topology/ContinuousFunction/Bounded.lean | 880 | 886 | theorem norm_eq_of_nonempty [h : Nonempty α] : ‖f‖ = sInf { C : ℝ | ∀ x : α, ‖f x‖ ≤ C } := by |
obtain ⟨a⟩ := h
rw [norm_eq]
congr
ext
simp only [mem_setOf_eq, and_iff_right_iff_imp]
exact fun h' => le_trans (norm_nonneg (f a)) (h' a)
|
/-
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
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 399 | 402 | theorem ConvolutionExistsAt.integrable_swap (h : ConvolutionExistsAt f g x L μ) :
Integrable (fun t => L (f (x - t)) (g t)) μ := by |
convert h.comp_sub_left x
simp_rw [sub_sub_self]
|
/-
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.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 198 | 201 | theorem noncommProd_map [MonoidHomClass F α β] (s : Multiset α) (comm) (f : F) :
f (s.noncommProd comm) = (s.map f).noncommProd (Multiset.noncommProd_map_aux s comm f) := by |
induction s using Quotient.inductionOn
simpa using map_list_prod f _
|
/-
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 | 92 | 100 | theorem mapAccumr_mapAccumr₂ (f₁ : γ → σ₁ → σ₁ × ζ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr f₁ (mapAccumr₂ f₂ xs ys s₂).snd s₁)
= let m := mapAccumr₂ (fun x y s =>
let r₂ := f₂ x y s.snd
let r₁ := f₁ r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs ys (s₁, s₂)
(m.fst.fst,... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import M... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 484 | 505 | theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 := by |
refine measure_null_of_locally_null _ fun x _ => ?_
obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ :=
Measure.exists_isOpen_measure_lt_top ρ x
let s := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
refine ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm ?_ bot_le⟩
have ρs : ρ s ... |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 100 | 110 | theorem cosZeta_two_mul_nat' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) / (2 * k) / Gammaℂ (2 * k) *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [cosZeta_two_mul_nat hk hx]
congr 1
have : (2 * k)! = (2 * k) * Complex.Gamma (2 * k) := by
rw [(by { norm_cast; omega } : 2 * (k : ℂ) = ↑(2 * k - 1) + 1), Complex.Gamma_nat_eq_factorial,
← Nat.cast_add_one, ← Nat.cast_mul, ← Nat.factorial_succ, Nat.sub_add_cancel (by omega)]
simp_rw [this, Gammaℂ... |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 145 | 147 | theorem add_le_add_add_tsub : a + b ≤ a + c + (b - c) := by |
rw [add_assoc]
exact add_le_add_left le_add_tsub a
|
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