Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | rw [isUnramified_iff_card_stabilizer_eq_one]
obtain (e|e) := nat_card_stabilizer_eq_one_or_two k w <;> rw [e] <;> decide
open scoped Classical in
lemma card_stabilizer [IsGalois k K] :
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 935 | 939 |
/-
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.Ideal
/-!
# Ideal operations for Lie algebras
Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Lie ideals of `L`... | @[simp]
theorem map_comap_incl : map N.incl (comap N.incl N') = N ⊓ N' := by
rw [← toSubmodule_inj]
exact (N : Submodule R M).map_comap_subtype N'
variable [LieAlgebra R L] [LieModule R L M₂] (I J : LieIdeal R L)
/-- Given a Lie module `M` over a Lie algebra `L`, the set of Lie ideals of `L` acts on the set
of su... | Mathlib/Algebra/Lie/IdealOperations.lean | 62 | 81 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 807 | 808 | |
/-
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.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,513 | 1,526 | |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
We gather results abou... | rwa [← mem_primitiveRoots hpos] at h
private theorem isRoot_cyclotomic_iff' {n : ℕ} {K : Type*} [Field K] {μ : K} [NeZero (n : K)] :
IsRoot (cyclotomic n K) μ ↔ IsPrimitiveRoot μ n := by
-- in this proof, `o` stands for `orderOf μ`
| Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 63 | 67 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Option.Basic
import Batteries.Tactic.Congr
import Mathlib.Data.Set.Basic
import Mathlib.Tactic.Contrapose
/-!
# Partial Equivalences
In this file, ... | c ∈ f.trans g a ↔ ∃ b, b ∈ f a ∧ c ∈ g b :=
| Mathlib/Data/PEquiv.lean | 136 | 136 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | | f _ _ h _ => simp [bodd_bit, bits_append_bit _ _ h]
theorem div2_bits_eq_tail (n : ℕ) : n.div2.bits = n.bits.tail := by
induction n using Nat.binaryRec' with
| Mathlib/Data/Nat/Bits.lean | 290 | 293 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | theorem associated_gcd_right_iff [GCDMonoid α] {x y : α} : Associated y (gcd x y) ↔ y ∣ x :=
⟨fun hx => hx.dvd.trans (gcd_dvd_left x y),
fun hxy => associated_of_dvd_dvd (dvd_gcd hxy dvd_rfl) (gcd_dvd_right x y)⟩
theorem Irreducible.isUnit_gcd_iff [GCDMonoid α] {x y : α} (hx : Irreducible x) :
IsUnit (gcd x ... | Mathlib/Algebra/GCDMonoid/Basic.lean | 625 | 663 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | theorem ofReal_ne_zero {x : ℝ} : (x : K) ≠ 0 ↔ x ≠ 0 :=
| Mathlib/Analysis/RCLike/Basic.lean | 167 | 167 |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | theorem monic_of_d_eq_one' : (toPoly ⟨0, 0, 0, 1⟩).Monic :=
monic_of_d_eq_one rfl rfl rfl rfl
| Mathlib/Algebra/CubicDiscriminant.lean | 231 | 233 |
/-
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.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 324 | 325 | |
/-
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.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... |
alias ⟨ModEq.zsmul_cancel, _⟩ := zsmul_modEq_zsmul
| Mathlib/Algebra/ModEq.lean | 149 | 150 |
/-
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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | rintro ⟨⟩ j
exact (H j).2
rcases this with ⟨i, h | h⟩
· let M' := transvection (inr Unit.unit) (inl i) 1 * M
have hM' : M' (inr unit) (inr unit) ≠ 0 := by simpa [M', hM]
rcases exists_isTwoBlockDiagonal_of_ne_zero M' hM' with ⟨L, L', hLL'⟩
rw [Matrix.mul_assoc] at hLL'
refine ⟨L ++ [⟨inr... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 567 | 604 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar... | · exact ((g_smooth.continuous_fderiv (mod_cast le_top)).clm_apply continuous_const).mul
hf.continuous
· exact ((g_comp.fderiv ℝ).comp_left rfl).mul_right
/-!
### Step 3: construct the derivative using the line derivatives along a basis
-/
theorem ae_exists_fderiv_of_countable
(hf : LipschitzWith C f) {s... | Mathlib/Analysis/Calculus/Rademacher.lean | 239 | 253 |
/-
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
/-! # P... | theorem abs_rpow_le_exp_log_mul (x y : ℝ) : |x ^ y| ≤ exp (log x * y) := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 154 | 154 |
/-
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.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... |
/-- The only continuous ring homomorphisms from `ℂ` to `ℂ` are the identity and the complex
conjugation. -/
| Mathlib/Analysis/Complex/Basic.lean | 232 | 234 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# Results about operator n... | Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean | 289 | 298 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathlib.Algebra.Polyno... | @[simp]
| Mathlib/Algebra/Polynomial/Taylor.lean | 46 | 46 |
/-
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.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... |
namespace Function
| Mathlib/Algebra/Ring/Parity.lean | 304 | 305 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 3,081 | 3,083 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,622 | 1,624 | |
/-
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.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,495 | 1,499 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | rw [not_and_or] at h0
rw [sin_angle_add_of_inner_eq_zero h h0, div_mul_cancel₀]
rw [← mul_self_ne_zero, norm_add_sq_eq_norm_sq_add_norm_sq_real h]
refine (ne_of_lt ?_).symm
rcases h0 with (h0 | h0)
· exact Left.add_pos_of_pos_of_nonneg (mul_self_pos.2 (norm_ne_zero_iff.2 h0)) (mul_self_nonneg _)
· exact L... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 162 | 170 |
/-
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.Set.Image
import Mathlib.Topology.Bases
import Mathlib.Topology.Inseparable
import Mathlib.Topology.Compactness.Exterior
/-!
# Alexandrov-discrete to... |
end
| Mathlib/Topology/AlexandrovDiscrete.lean | 202 | 203 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... | theorem IsSeparatedMap.isClosed_eqLocus (sep : IsSeparatedMap f) (he : f ∘ g₁ = f ∘ g₂) :
IsClosed {a | g₁ a = g₂ a} :=
let g : A → f.Pullback f := fun a ↦ ⟨⟨g₁ a, g₂ a⟩, congr_fun he a⟩
| Mathlib/Topology/SeparatedMap.lean | 181 | 183 |
/-
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.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,303 | 1,308 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.Curry
import Mathlib.Data.Set.Countable
/-!
# Filters with countable intersection property
In this file we define `CountableInterFilte... | `Filter.univ_sets` and `Filter.inter_sets` from the countable intersection property. -/
def Filter.ofCountableInter (l : Set (Set α))
(hl : ∀ S : Set (Set α), S.Countable → S ⊆ l → ⋂₀ S ∈ l)
(h_mono : ∀ s t, s ∈ l → s ⊆ t → t ∈ l) : Filter α where
sets := l
univ_sets := @sInter_empty α ▸ hl _ countable_empt... | Mathlib/Order/Filter/CountableInter.lean | 116 | 121 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Complement
import Mathlib.GroupTheory.Sylow
/-!
# The Transfer Homomorphism
In this file we construct the transfer homomorphism.
## Ma... | apply_symm_apply]
lemma transferTransversal_apply'' (q : orbitRel.Quotient (zpowers g) (G ⧸ H))
(k : ZMod (minimalPeriod (g • ·) q.out)) :
↑((g • transferTransversal H g).2.leftQuotientEquiv (g ^ (cast k : ℤ) • q.out)) =
if k = 0 then g ^ minimalPeriod (g • ·) q.out * q.out.out
else g ^ (cast k... | Mathlib/GroupTheory/Transfer.lean | 118 | 139 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 2,158 | 2,160 | |
/-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.Da... | hfg.sum_comp_perm_smul_le_sum_smul hσ]
/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
| Mathlib/Algebra/Order/Rearrangement.lean | 293 | 295 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | for a version that works for non-injective maps. -/
theorem completeSpace_congr {e : α ≃ β} (he : IsUniformEmbedding e) :
CompleteSpace α ↔ CompleteSpace β :=
he.completeSpace_congr e.surjective
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 328 | 331 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | instance instAddCommMonoid [AddCommMonoid M] : AddCommMonoid (α →₀ M) :=
fast_instance% DFunLike.coe_injective.addCommMonoid
DFunLike.coe coe_zero coe_add (fun _ _ => rfl)
| Mathlib/Data/Finsupp/Defs.lean | 576 | 579 |
/-
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.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | dvd_trans h.orderOf_mul_dvd_lcm (lcm_dvd_mul _ _)
@[to_additive addOrderOf_add_eq_mul_addOrderOf_of_coprime]
theorem orderOf_mul_eq_mul_orderOf_of_coprime (h : Commute x y)
(hco : (orderOf x).Coprime (orderOf y)) : orderOf (x * y) = orderOf x * orderOf y := by
| Mathlib/GroupTheory/OrderOfElement.lean | 420 | 424 |
/-
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
/-!
# Rotations by oriented angles.
This... | apply LinearIsometryEquiv.toLinearEquiv_injective
apply LinearEquiv.toLinearMap_injective
apply (o.basisRightAngleRotation x hx).ext
intro i
symm
fin_cases i
· simp
have : o.oangle (J x) (f (J x)) = o.oangle x (f x) := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 338 | 345 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | theorem toSemiring_injective :
Function.Injective (@toSemiring R) := by
intro _ _ h
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| Mathlib/Algebra/Ring/Ext.lean | 339 | 344 |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | | n, k+1, s => by
rw [take]
cases h : destruct s with
| none =>
simp [destruct_eq_none h]
| some a =>
match a with
| (x, r) =>
| Mathlib/Data/Seq/Seq.lean | 657 | 664 |
/-
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.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 422 | 423 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | rw [TopologicalSpace.ext_iff, compl_surjective.forall]
simp only [@isOpen_compl_iff _ _ t₁, @isOpen_compl_iff _ _ t₂]
alias ⟨_, TopologicalSpace.ext_isClosed⟩ := TopologicalSpace.ext_iff_isClosed
| Mathlib/Topology/Basic.lean | 119 | 123 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... | simp only [← mul_assoc, ← pow_add, ← pow_mul]
cases n
| Mathlib/Algebra/Order/Ring/Basic.lean | 233 | 234 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
/-!
# The initial algebra of a multivariate qpf is again a qpf.
Fo... | abs_repr := by
intro α
apply Quot.ind
intro a
apply Quot.sound
apply wrepr_equiv
abs_map := by
intro α β g x
| Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 287 | 294 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 878 | 881 | |
/-
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.Finset.Pi
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite.Basic
/-!
# Fintype instances for pi types
-/
assert_not_exists Order... | (hts : t ⊆ s i) : piFinset (Function.update s i t) = {f ∈ piFinset s | f i ∈ t} := by
ext f
simp only [mem_piFinset, mem_filter]
refine ⟨fun h ↦ ?_, fun h j ↦ ?_⟩
· have := by simpa using h i
refine ⟨fun j ↦ ?_, this⟩
obtain rfl | hji := eq_or_ne j i
· exact hts this
· simpa [hji] using h j
... | Mathlib/Data/Fintype/Pi.lean | 114 | 126 |
/-
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.Order.Sub.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
/-!
# Lemma about subtraction in ordered monoids with a top element a... | | (x : α), (y : α) => by simp only [← coe_sub, map_coe, h]
end
variable [Add α] [LE α] [OrderBot α] [Sub α] [OrderedSub α]
| Mathlib/Algebra/Order/Sub/WithTop.lean | 64 | 68 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... | (Cardinal.sum fun l => Cardinal.lift.{v} #(L.Functions l)) +
Cardinal.sum fun l => Cardinal.lift.{u} #(L.Relations l) := by
simp only [card, mk_sum, mk_sigma, lift_sum]
| Mathlib/ModelTheory/Basic.lean | 99 | 101 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.P... | { pt := s.pt
ι := _ } m ?_
rintro ⟨j, k⟩
dsimp
| Mathlib/CategoryTheory/Limits/Fubini.lean | 288 | 291 |
/-
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, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 582 | 583 | |
/-
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.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | angle. -/
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 253 | 254 |
/-
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.Split
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Construction of the inverse functor of the Dold-Kan equivalence
... | unfold mapMono
rw [Ne] at h₁
split_ifs
rfl
variable {K K'}
| Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 112 | 117 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | Expressions are represented as a list of terms of the form `e = n • x`,
where `n : ℤ` and `x` is an arbitrary element of the additive commutative monoid or group.
We explicitly track the `Expr` forms of `e` and `n`, even though they could be reconstructed,
| Mathlib/Tactic/Abel.lean | 137 | 139 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.CategoryTheory.Sites.Canonical
/-!
# Grothendieck Topology and Sheaves on the Category of Types
In this file we define a Grothendieck topology on the category ... | funext α
ext S
refine ⟨fun hs x => ?_, fun hs β f => Presieve.isSheaf_yoneda' _ fun y => hs _⟩
by_contra hsx
have : (fun _ => ULift.up true) = fun _ => ULift.up false :=
(hs PUnit fun _ => x).isSeparatedFor.ext
fun β f hf => funext fun y => hsx.elim <| S.2 hf fun _ => y
simp [funext_iff] at this
... | Mathlib/CategoryTheory/Sites/Types.lean | 182 | 193 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Michael Stoll
-/
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.Data.Complex.FiniteDimensional
/-!
# L-series
Gi... | simp [LSeries, LSeries.term]
section delta
| Mathlib/NumberTheory/LSeries/Basic.lean | 269 | 272 |
/-
Copyright (c) 2024 Yaël Dillies, Kin Yau James Wong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Kin Yau James Wong, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Probability.Kernel.Composition.MeasureCompProd
... | simp [this, (measurableEmbedding_prodMk_left x).comap_apply, hx, Set.singleton_prod]
lemma IsCondKernel.isProbabilityMeasure [MeasurableSingletonClass α] {a : α} (ha : ρ.fst {a} ≠ 0) :
IsProbabilityMeasure (ρCond a) := by
constructor
rw [IsCondKernel.apply_of_ne_zero _ _ ha, prod_univ, ← Measure.fst_apply
... | Mathlib/Probability/Kernel/Disintegration/Basic.lean | 95 | 128 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | open scoped Classical in
@[simp]
theorem piecewise_same (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) :
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 214 | 216 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | obtain ⟨n, rfl⟩ := hf.2 ha hb
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 797 | 797 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Logic.Equiv.Functor
import Mathlib.RingTheory.FreeRing
/-... | @[simp, norm_cast]
protected theorem coe_add (x y : FreeRing α) : ↑(x + y) = (x : FreeCommRing α) + y :=
| Mathlib/RingTheory/FreeCommRing.lean | 341 | 342 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 1,266 | 1,269 | |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Algebra.Lie.... | skewAdjointMatricesLieSubalgebra <| indefiniteDiagonal p q R
/-- A matrix for transforming the indefinite diagonal bilinear form into the definite one, provided
| Mathlib/Algebra/Lie/Classical.lean | 154 | 156 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 1,171 | 1,172 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | congr_arg ((↑) : ℕ → ℤ) (by simp [Nat.shiftLeft_eq])
| Mathlib/Data/Int/Bitwise.lean | 412 | 413 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... |
@[simp]
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 217 | 218 |
/-
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.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... |
nonrec theorem MellinConvergent.div_const {f : ℝ → ℂ} {s : ℂ} (hf : MellinConvergent f s) (a : ℂ) :
MellinConvergent (fun t => f t / a) s := by
simpa only [MellinConvergent, smul_eq_mul, ← mul_div_assoc] using hf.div_const a
| Mathlib/Analysis/MellinTransform.lean | 53 | 56 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | rw [← integrableOn_iff_integrable_of_support_subset A]
apply Integrable.smul_of_top_right hif
exact memLp_top_of_bound hig.aestronglyMeasurable.restrict
| Mathlib/MeasureTheory/Integral/Average.lean | 783 | 785 |
/-
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.Combinatorics.SimpleGraph.Prod
import Mathlib.Data.Fin.SuccPred
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Relation
import Mathlib.Tact... | fun a b => by
rw [reachable_iff_reflTransGen, ← reflTransGen_swap]
exact
reflTransGen_of_pred _ (fun c hc => Or.inl <| pred_covBy_of_not_isMin hc.1.not_isMin)
fun c hc => Or.inr <| pred_covBy_of_not_isMin hc.1.not_isMin
| Mathlib/Combinatorics/SimpleGraph/Hasse.lean | 85 | 90 |
/-
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
/-!
# Positive operators
In this file we define positive operators in a Hilbert space. We follow Bourbaki's ch... | refine ⟨?_, ⟨_, h_anti⟩⟩
have _inst := h_anti.completeSpace_range_clm
rw [Submodule.topologicalClosure_eq_top_iff, Submodule.eq_bot_iff]
intro x hx
have : ‖x‖ ^ 2 * c = 0 := le_antisymm (by simpa only [hx (f x) ⟨x, rfl⟩, norm_zero] using h x)
| Mathlib/Analysis/InnerProductSpace/Positive.lean | 119 | 123 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | have hgcd_ne_zero : x.gcd y ≠ 0 := gcd_ne_zero_left hx
have hlcm_ne_zero : x.lcm y ≠ 0 := lcm_ne_zero hx hy
have hfi_zero : ∀ {i}, f (i ^ 0) = 1 := by
intro i; rw [Nat.pow_zero, hf.1]
iterate 4 rw [hf.multiplicative_factorization f (by assumption),
| Mathlib/NumberTheory/ArithmeticFunction.lean | 735 | 739 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | @[reassoc (attr := simp)]
lemma inr_desc :
inr φ ≫ desc φ α hα = α := by aesop_cat
| Mathlib/Algebra/Homology/HomotopyCofiber.lean | 316 | 318 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | empty_subset _
@[simp] theorem surjOn_empty_iff : SurjOn f ∅ t ↔ t = ∅ := by
simp [SurjOn, subset_empty_iff]
@[simp] lemma surjOn_singleton : SurjOn f s {b} ↔ b ∈ f '' s := singleton_subset_iff
| Mathlib/Data/Set/Function.lean | 451 | 457 |
/-
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
/-!
# Ordered Subtraction
This file proves l... | rw [add_comm a, add_comm (a - c)]
exact add_tsub_le_assoc
| Mathlib/Algebra/Order/Sub/Defs.lean | 155 | 157 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... | ext ⟨x, y⟩
simp [and_left_comm, eq_comm]
theorem singleton_sigma_singleton {a : ∀ i, α i} :
(({i} : Set ι).sigma fun i ↦ ({a i} : Set (α i))) = {⟨i, a i⟩} := by
rw [sigma_singleton, image_singleton]
@[simp]
| Mathlib/Data/Set/Sigma.lean | 100 | 107 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | inter_comm]
theorem restrict_eq_self_of_ae_mem {_m0 : MeasurableSpace α} ⦃s : Set α⦄ ⦃μ : Measure α⦄
| Mathlib/MeasureTheory/Measure/Restrict.lean | 298 | 300 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Order.Hom.CompleteLattice
import Mathlib.Topology.Compactness.Bases
import Mathlib.Topology.ContinuousMap.Basic
impor... | map_inf' _ _ := rfl
map_top' := rfl
@[simp]
theorem comap_id : comap (ContinuousMap.id α) = FrameHom.id _ :=
FrameHom.ext fun _ => ext rfl
theorem comap_mono (f : C(α, β)) {s t : Opens β} (h : s ≤ t) : comap f s ≤ comap f t :=
OrderHomClass.mono (comap f) h
@[simp]
theorem coe_comap (f : C(α, β)) (U : Opens ... | Mathlib/Topology/Sets/Opens.lean | 343 | 357 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | · exact hx i
· intro j _ hji
rw [coeff_C_mul, coeff_X_pow, if_neg (hk.injective.ne hji.symm), mul_zero]
· exact fun hi => (hi (mem_univ i)).elim
· intro i
by_cases hi : ∃ i₀, Fin.castSucc i₀ = i
· obtain ⟨i, rfl⟩ := hi
rw [Fin.strictMono_castSucc.injective.exten... | Mathlib/Algebra/Polynomial/EraseLead.lean | 378 | 421 |
/-
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.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... |
section Zero
variable [∀ x, Zero (E x)]
| Mathlib/Topology/FiberBundle/Trivialization.lean | 214 | 217 |
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | theorem trace_kroneckerMapBilinear [CommSemiring R] [Fintype m] [Fintype n] [AddCommMonoid α]
[AddCommMonoid β] [AddCommMonoid γ] [Module R α] [Module R β] [Module R γ]
(f : α →ₗ[R] β →ₗ[R] γ) (A : Matrix m m α) (B : Matrix n n β) :
trace (kroneckerMapBilinear f A B) = f (trace A) (trace B) := by
simp_rw ... | Mathlib/Data/Matrix/Kronecker.lean | 209 | 219 |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu, Jack McKoen
-/
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.Algebra.Ring.Subring.Pointwise
impor... | · rintro h x (hx : A.valuation x = 1)
apply_fun A.mapOfLE B h at hx
| Mathlib/RingTheory/Valuation/ValuationSubring.lean | 461 | 462 |
/-
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
/-!
# Topological star (sub)algebras
A topological... | rintro ⟨y, rfl⟩
exact y.prop, fun hy => ⟨⟨y, hy⟩, rfl⟩⟩
refine Continuous.ext_on this hφ hψ ?_
rintro _ ⟨x, rfl⟩
simpa only using DFunLike.congr_fun h x
theorem _root_.StarAlgHomClass.ext_topologicalClosure [T2Space B] {F : Type*}
{S : StarSubalgebra R A} [FunLike F S.topologicalClosure B... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 146 | 163 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | (a ^ p + b ^ p) ^ (1 / p) ≤ a + b := by
lift a to NNReal using ha
lift b to NNReal using hb
exact_mod_cast NNReal.rpow_add_rpow_le_add a b hp1
| Mathlib/Analysis/MeanInequalitiesPow.lean | 196 | 200 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Field.Defs
/-!
# Linear ordered (semi)fields
A ... | Mathlib/Algebra/Order/Field/Defs.lean | 47 | 49 | |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Operator.LinearIsometry
import Mathlib.LinearAlgebra.Basis.VectorSpace
/-!
# Conformal Linear ... | exact ⟨c * c', mul_ne_zero hc hc', li, smul_smul _ _ _⟩
theorem isConformalMap_const_smul (hc : c ≠ 0) : IsConformalMap (c • id R M) :=
isConformalMap_id.smul hc
| Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 62 | 65 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,324 | 1,328 | |
/-
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, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.Commo... | Mathlib/Algebra/Divisibility/Basic.lean | 242 | 246 | |
/-
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.GeomSum
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.NoZeroSMulDiviso... | IsNilpotent (x - y) := by
rw [← neg_right_iff] at h_comm
rw [← isNilpotent_neg_iff] at hy
rw [sub_eq_add_neg]
exact h_comm.isNilpotent_add hx hy
end Ring
end Commute
section CommSemiring
variable [CommSemiring R] {x y : R}
| Mathlib/RingTheory/Nilpotent/Basic.lean | 196 | 208 |
/-
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.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | calc
f m ≤ f (m ⊔ n) := hf le_sup_left
| Mathlib/Order/Lattice.lean | 235 | 236 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... |
@[reassoc (attr := simp)]
theorem inv_hom_id_tensor' {V W X Y Z : C} (f : V ⟶ W) [IsIso f] (g : X ⟶ Y) (h : Y ⟶ Z) :
| Mathlib/CategoryTheory/Monoidal/Category.lean | 666 | 668 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Nat.Prime.Basic
/-!
# `norm_num` extensions on natural numbers
This... |
theorem isNat_minFac_3 : {n n' : ℕ} → (k : ℕ) →
| Mathlib/Tactic/NormNum/Prime.lean | 100 | 101 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.Group.Subgroup.MulOppositeLemmas
import Mathlib.Algebra.Gr... | S * N = N * S := by
rw [← iUnion_mul_left_image, ← iUnion_mul_right_image]
| Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 232 | 233 |
/-
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.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... |
theorem pred_le : ∀ a : α, pred a ≤ a :=
| Mathlib/Order/SuccPred/Basic.lean | 566 | 567 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 827 | 829 | |
/-
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.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousMap.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
/-!
# Homo... | prop' : ∀ t, P ⟨fun x => toFun (t, x),
Continuous.comp continuous_toFun (continuous_const.prodMk continuous_id')⟩
| Mathlib/Topology/Homotopy/Basic.lean | 373 | 374 |
/-
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.Group.Basic
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Data.Nat.Basic
/-!
# Partial predecessor and partial subtraction on the natur... | theorem ppred_eq_some {m : ℕ} : ∀ {n}, ppred n = some m ↔ succ m = n
| 0 => by constructor <;> intro h <;> contradiction
| n + 1 => by constructor <;> intro h <;> injection h <;> subst m <;> rfl
| Mathlib/Data/Nat/PSub.lean | 57 | 59 |
/-
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
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | rw [← map_prod_eq_map₂ n, ← map_prod_eq_map₂ n', map₂_map_left, map₂_map_right,
← map_prod_eq_map₂, ← map_prod_eq_map₂, ← prod_assoc, map_map]
simp only [h_assoc, Function.comp_def, Equiv.prodAssoc_apply]
| Mathlib/Order/Filter/NAry.lean | 172 | 174 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,028 | 1,029 | |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph,
if present. Any edges incident to the deleted vertices are deleted as well. -/
abbrev deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph :=
G'.induce (G'.verts \ s)
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,206 | 1,210 |
/-
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.Nat
import Mathlib.Data.PNat.Defs
/-!
# Finite intervals of positive naturals
This file proves that `ℕ+` is a `LocallyFiniteOrder` ... |
theorem card_fintype_uIcc : Fintype.card (Set.uIcc a b) = (b - a : ℤ).natAbs + 1 := by
rw [← card_uIcc, Fintype.card_ofFinset]
end PNat
| Mathlib/Data/PNat/Interval.lean | 85 | 90 |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... |
end Polynomial
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 569 | 570 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 2,728 | 2,748 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.StronglyMeasura... | map_smul' := lpMeasToLpTrim_smul hm
norm_map' := lpMeasSubgroupToLpTrim_norm_map hm
variable {F 𝕜 p μ}
instance [hm : Fact (m ≤ m0)] [CompleteSpace F] [hp : Fact (1 ≤ p)] :
CompleteSpace (lpMeasSubgroup F m p μ) := by
rw [(lpMeasSubgroupToLpTrimIso F p μ hm.elim).completeSpace_iff]; infer_instance
-- For ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 413 | 423 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... |
@[simp]
theorem C_add_two : ∀ n, C R (n + 2) = X * C R (n + 1) - C R n
| (k : ℕ) => C.eq_3 R k
| -(k + 1 : ℕ) => by linear_combination (norm := (simp [Int.negSucc_eq]; ring_nf)) C.eq_4 R k
theorem C_add_one (n : ℤ) : C R (n + 1) = X * C R n - C R (n - 1) := by
linear_combination (norm := ring_nf) C_add_two R (n... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 296 | 303 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Combinatorics.Hall.Basic
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Projectivization.Constructions
/-!
# Configurations of Points ... |
theorem two_lt_pointCount [Finite P] [Finite L] (l : L) : 2 < pointCount P l := by
simpa only [pointCount_eq P l, Nat.succ_lt_succ_iff] using one_lt_order P L
variable (L)
theorem card_points [Fintype P] [Finite L] : Fintype.card P = order P L ^ 2 + order P L + 1 := by
| Mathlib/Combinatorics/Configuration.lean | 424 | 430 |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... |
theorem leadingCoeff_comp (hq : natDegree q ≠ 0) :
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 366 | 367 |
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