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) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 680 | 681 | |
/-
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... | · constructor
· exact t_l_valid.bal
| Mathlib/Data/Ordmap/Ordset.lean | 562 | 563 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... |
theorem TransGen.mono {p : α → α → Prop} :
(∀ a b, r a b → p a b) → TransGen r a b → TransGen p a b :=
TransGen.lift id
lemma transGen_minimal {r' : α → α → Prop} (hr' : Transitive r') (h : ∀ x y, r x y → r' x y)
| Mathlib/Logic/Relation.lean | 494 | 499 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
/-!
# Some results on tensor product of s... | @[simp]
theorem comm_trans_lTensorOne :
(TensorProduct.comm R _ _).trans M.lTensorOne = M.rTensorOne := by
refine LinearEquiv.toLinearMap_injective <| TensorProduct.ext' fun m r ↦ ?_
| Mathlib/LinearAlgebra/TensorProduct/Submodule.lean | 234 | 237 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
/-!
# Additional lemmas about sum types
Most of the former contents ... | {x : γ} : update f (inl i) x ∘ inl = update (f ∘ inl) i x :=
update_comp_eq_of_injective _ inl_injective _ _
| Mathlib/Data/Sum/Basic.lean | 78 | 79 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... | Mathlib/CategoryTheory/Abelian/Exact.lean | 363 | 368 | |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | lemma pos_iff_pos_of_smul_pos [PosSMulReflectLT α β] [SMulPosReflectLT α β] (hab : 0 < a • b) :
0 < a ↔ 0 < b :=
⟨pos_of_smul_pos_left hab ∘ le_of_lt, pos_of_smul_pos_right hab ∘ le_of_lt⟩
| Mathlib/Algebra/Order/Module/Defs.lean | 515 | 518 |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | instance instPosSMulMono [∀ i, PosSMulMono α (β i)] : PosSMulMono α (∀ i, β i) where
elim _a ha _b₁ _b₂ hb i := smul_le_smul_of_nonneg_left (hb i) ha
| Mathlib/Algebra/Order/Module/Defs.lean | 1,014 | 1,015 |
/-
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.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathl... | namespace HasStrictFDerivAt
section Complemented
/-!
### Case of a complemented kernel
In this section we prove the following version of the implicit function theorem. Consider a map
`f : E → F` and a point `a : E` such that `f` is strictly differentiable at `a`, its derivative `f'`
is surjective and the kernel of `... | Mathlib/Analysis/Calculus/Implicit.lean | 201 | 214 |
/-
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.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | rintro ⟨p, hp, rfl⟩
ext1
simp only
conv_rhs => rw [← eraseLead_add_C_mul_X_pow p]
simp [Monic.def.1 hp, add_comm]
right_inv := by
| Mathlib/RingTheory/Polynomial/Basic.lean | 186 | 191 |
/-
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.Defs
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topol... | algebraMap := (UniformSpace.Completion.coeRingHom : A →+* Completion A).comp (algebraMap R A)
commutes' := fun r x =>
Completion.induction_on x (isClosed_eq (continuous_mul_left _) (continuous_mul_right _))
fun a => by
simpa only [coe_mul] using congr_arg ((↑) : A → Completion A) (Algebra.commutes r... | Mathlib/Topology/Algebra/UniformRing.lean | 195 | 200 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 1,157 | 1,168 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl, Reid Barton
-/
import Mathlib.CategoryTheory.Category.Init
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Tactic.PPWithUniv
import... | instance : Epi f where
left_cancellation _ _ _ := Subsingleton.elim _ _
end
end
| Mathlib/CategoryTheory/Category/Basic.lean | 341 | 347 |
/-
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, Simon Hudon, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | -/
class AddGroup (A : Type u) extends SubNegMonoid A where
| Mathlib/Algebra/Group/Defs.lean | 1,067 | 1,068 |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... |
@[simp]
theorem nil_toMultiset : nil.toMultiset = (0 : Multiset α) :=
rfl
@[simp]
| Mathlib/Data/List/Cycle.lean | 604 | 609 |
/-
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... | toFun a₂ :=
haveI := Classical.decEq β
| Mathlib/Data/Finsupp/Defs.lean | 369 | 370 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Logic.Basic
/-!
# Characteristic zero
A ring `R` is called of characteristic zero if every natural number `n` is n... | @[norm_cast]
| Mathlib/Algebra/CharZero/Defs.lean | 79 | 79 |
/-
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.Algebra.Hom
import Mathlib.Algebra.GroupWithZero.Action.Prod
/-!
# Morphisms of non-unital algebras
This file defines morphisms between two types, ... | rfl
@[simp, norm_cast]
theorem coe_to_distribMulActionHom (f : A →ₛₙₐ[φ] B) : ⇑(f : A →ₑ+[φ] B) = f :=
| Mathlib/Algebra/Algebra/NonUnitalHom.lean | 211 | 214 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | Mathlib/Algebra/Order/Ring/Abs.lean | 74 | 74 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | have : ∀ x : X, ({x} : Set X) ∈ 𝓝 x := by simp [nhds_discrete]
rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with ⟨t, _, hst⟩
simp only [← t.set_biUnion_coe, biUnion_of_singleton] at hst
| Mathlib/Topology/Compactness/Compact.lean | 468 | 470 |
/-
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.Data.Complex.Trigonometric
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
/-!
# Multiple angle formulas in terms of... | evaluates on `2 * cos θ` to the value `sin ((n + 1) * θ) / sin θ`. -/
@[simp]
theorem S_two_mul_complex_cos (n : ℤ) : (S ℂ n).eval (2 * cos θ) * sin θ = sin ((n + 1) * θ) := by
simp [S_eq_U_comp_half_mul_X]
/-- The `n`-th Chebyshev polynomial of the first kind evaluates on `cosh θ` to the
value `cosh (n * θ)`. -/
@[... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 92 | 105 |
/-
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.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | rcases exists_pos_mul_lt hε a with ⟨ε', hε', ha⟩
rcases h ε' hε' with ⟨r, hr, H⟩
exact ⟨r, hr, fun c π hπ hπp => (H c π hπ hπp).trans ha.le⟩
theorem integrable_iff_cauchy [CompleteSpace F] :
| Mathlib/Analysis/BoxIntegral/Basic.lean | 194 | 198 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-! # Power funct... | · rwa [← le_div_iff₀' (Nat.cast_pos.2 h₀.bot_lt)]
lemma pow_cpow_ofNat_inv {x : ℂ} {n : ℕ} [n.AtLeastTwo] (hlt : -(π / OfNat.ofNat n) < x.arg)
(hle : x.arg ≤ π / OfNat.ofNat n) :
(x ^ ofNat(n)) ^ ((OfNat.ofNat n : ℂ)⁻¹) = x :=
pow_cpow_nat_inv (NeZero.ne n) hlt hle
/-- See also `Complex.pow_cpow_ofNat_inv... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 175 | 182 |
/-
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.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,299 | 2,301 | |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,360 | 1,382 | |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# The layer cake formula / Cavalieri's principle / tail probability formula
In this file we prove the f... | ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
/- We will reduce to the sigma-finite case, after excluding two easy cases where the result
is more or less obvious. -/
have f_nonneg : ∀ ω, 0 ≤ f ω := fun ω ↦ f_nn ω
-- trivial case where ... | Mathlib/MeasureTheory/Integral/Layercake.lean | 195 | 381 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 816 | 820 | |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom =
(α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom := by
simp [← cancel_epi ((α_ f g h).hom ▷ i)]
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 267 | 270 |
/-
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... | @[simp]
theorem toIocMod_toReal (θ : Angle) : toIocMod two_pi_pos (-π) θ.toReal = θ.toReal := by
induction θ using Real.Angle.induction_on
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 474 | 476 |
/-
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 | 2,864 | 2,866 | |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | lift_sepDegree_eq_of_equiv _ _ _ ((botEquiv E K).restrictScalars F)
theorem lift_insepDegree_bot' : Cardinal.lift.{v} (insepDegree F (⊥ : IntermediateField E K)) =
| Mathlib/FieldTheory/SeparableClosure.lean | 353 | 355 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | lemma fg_of_isNoetherianRing [hR : IsNoetherianRing R] (hS : S ≤ R⁰) (I : FractionalIdeal S P) :
FG I.coeToSubmodule := by
have := hR.noetherian I.num
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 676 | 678 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | theorem IsEquivalent.isTheta_symm (h : u ~[l] v) : v =Θ[l] u :=
⟨h.isBigO_symm, h.isBigO⟩
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 89 | 91 |
/-
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... | attribute [local instance] Subtype.measureSpace
| Mathlib/MeasureTheory/Measure/Restrict.lean | 750 | 751 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... | /-- The distance from the orthocenter to the reflection of the
circumcenter in a side equals the circumradius, variant using a
`Finset`. -/
theorem dist_orthocenter_reflection_circumcenter_finset (t : Triangle ℝ P) {i₁ i₂ : Fin 3}
(h : i₁ ≠ i₂) :
dist t.orthocenter
(reflection (affineSpan ℝ (t.points ''... | Mathlib/Geometry/Euclidean/MongePoint.lean | 392 | 423 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
/-!
# Intersecting families
This file defines intersecting families and proves their basic proper... | classical
refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
rw [Fintype.card, ← this, Nat.two_mul, card_disjUnio... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 155 | 160 |
/-
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, Sander Dahmen, Kim Morrison
-/
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAl... | finite_def.2 (b.nonempty_fintype_index_of_rank_lt_aleph0 h)
end
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 167 | 170 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | (prod.associator (W ⨯ X) Y Z).hom ≫ (prod.associator W X (Y ⨯ Z)).hom := by
simp
@[reassoc]
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 896 | 899 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | Mathlib/Data/Finset/NAry.lean | 615 | 618 | |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.Tactic.NormNum.DivMod
import Mathlib.Tactic.NormNum.PowMod
import Mathlib.Tactic.Re... |
lemma CharP.isInt_of_mod {e' r : ℤ} {α : Type*} [Ring α] {n n' : ℕ} (inst : CharP α n) {e : α}
(he : IsInt e e') (hn : IsNat n n') (h₂ : IsInt (e' % n') r) : IsInt e r :=
⟨by rw [he.out, CharP.intCast_eq_intCast_mod α n, show n = n' from hn.out, h₂.out, Int.cast_id]⟩
| Mathlib/Tactic/ReduceModChar.lean | 47 | 51 |
/-
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.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... | simp only [Filter.mem_map, mem_ae_iff, ← hf.measure_preimage (hg hs.measurableSet).compl,
preimage_comp, preimage_compl]
theorem aeconst_preimage {f : α → β} (hf : MeasurePreserving f μa μb) {s : Set β}
(hs : NullMeasurableSet s μb) :
Filter.EventuallyConst (f ⁻¹' s) (ae μa) ↔ Filter.EventuallyConst ... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 152 | 159 |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 487 | 501 | |
/-
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... | Mathlib/Data/Finsupp/Defs.lean | 1,122 | 1,123 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | theorem image_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ) :
h '' Metric.closedBall x r = Metric.closedBall (h x) r := by
| Mathlib/Topology/MetricSpace/Isometry.lean | 599 | 600 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | theorem le_ncard_diff (s t : Set α) (hs : s.Finite := by toFinite_tac) :
t.ncard - s.ncard ≤ (t \ s).ncard :=
tsub_le_iff_left.mpr (by rw [add_comm]; apply ncard_le_ncard_diff_add_ncard _ _ hs)
| Mathlib/Data/Set/Card.lean | 893 | 896 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... |
end
| Mathlib/Data/Set/Lattice.lean | 473 | 474 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | ext i
dsimp
rw [← ConcreteCategory.comp_apply, Multiequalizer.lift_ι]
rfl
variable (J)
include instCC
/-- `P⁺⁺` is always a sheaf. -/
theorem isSheaf_plus_plus (P : Cᵒᵖ ⥤ D) : Presheaf.IsSheaf J (J.plusObj (J.plusObj P)) := by
apply isSheaf_of_sep
intro X S x y
apply sep
end
end Plus
variabl... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 409 | 434 |
/-
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.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | theorem natDegree_hasseDeriv [NoZeroSMulDivisors ℕ R] (p : R[X]) (n : ℕ) :
natDegree (hasseDeriv n p) = natDegree p - n := by
rcases lt_or_le p.natDegree n with hn | hn
· simpa [hasseDeriv_eq_zero_of_lt_natDegree, hn] using (tsub_eq_zero_of_le hn.le).symm
· refine map_natDegree_eq_sub ?_ ?_
· exact fun h ... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 192 | 207 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 427 | 428 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... |
theorem exists_isCompact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
simp_rw [Ne, ← measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact,
ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and]
| Mathlib/MeasureTheory/Measure/Regular.lean | 991 | 994 |
/-
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.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | theorem HasLineDerivWithinAt.congr_mono (h : HasLineDerivWithinAt 𝕜 f f' s x v) (ht : EqOn f₁ f t)
(hx : f₁ x = f x) (h₁ : t ⊆ s) : HasLineDerivWithinAt 𝕜 f₁ f' t x v :=
HasDerivWithinAt.congr_mono h (fun _ hy ↦ ht hy) (by simpa using hx) (preimage_mono h₁)
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 177 | 179 |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on ℕ
This file defines a simple dista... | _ = n - m + (m + k - (n + k)) := by rw [@add_tsub_add_eq_tsub_right]
| Mathlib/Data/Nat/Dist.lean | 60 | 60 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | ((chain'_map f).1 p).imp H
theorem chain'_map_of_chain' {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
| Mathlib/Data/List/Chain.lean | 216 | 218 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Jacobson.Ideal
import Mathlib.RingTheory.Ideal.Quo... | variable (I M)
instance : IsHausdorff I (AdicCompletion I M) where
haus' x h := ext fun n ↦ by
refine smul_induction_on (SModEq.zero.1 <| h n) (fun r hr x _ ↦ ?_) (fun x y hx hy ↦ ?_)
| Mathlib/RingTheory/AdicCompletion/Basic.lean | 324 | 328 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
/-!
# Congruence subgroups
This defines congruence subgroups of `SL(2, ℤ)` such as `Γ(N)`, `Γ₀(N)` and `Γ₁(N)... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 248 | 253 | |
/-
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.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | simp only [Set.mem_singleton_iff, Set.mem_insert_iff, mul_eq_left, mul_eq_right,
mul_eq_one_iff_eq_inv, inv_eq_self_of_orderOf_eq_two hy, not_or]
aesop
@[to_additive]
lemma mul_not_mem_of_exponent_two (h : Monoid.exponent G = 2) {x y : G}
(hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : x * y ∉ ({x, y, 1} : Set G... | Mathlib/GroupTheory/Exponent.lean | 648 | 656 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,617 | 1,619 | |
/-
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.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,591 | 1,594 | |
/-
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
import Mathlib.Data.Complex.Trig... |
lemma summable_exp_neg_nat : Summable fun n : ℕ ↦ exp (-n) := by
simpa only [mul_neg_one] using summable_exp_nat_mul_iff.mpr neg_one_lt_zero
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 427 | 429 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# With Density
For an s-finite kernel `κ : K... | rw [withDensity, dif_pos hf]
rfl
protected theorem withDensity_apply' (κ : Kernel α β) [IsSFiniteKernel κ]
(hf : Measurable (Function.uncurry f)) (a : α) (s : Set β) :
withDensity κ f a s = ∫⁻ b in s, f a b ∂κ a := by
| Mathlib/Probability/Kernel/WithDensity.lean | 60 | 65 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 518 | 521 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | show sInf { b : ℝ≥0∞ | 1 ≤ 0 * b } = ∞ by simp
| Mathlib/Data/ENNReal/Inv.lean | 46 | 47 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
/-!
# Bilinear form
This file defines the conversio... | h.eq_zero_of_ortho fun y => by simpa only [toBilin'_apply'] using hx y
@[simp]
theorem _root_.Matrix.nondegenerate_toBilin'_iff {M : Matrix ι ι R₂} :
M.toBilin'.Nondegenerate ↔ M.Nondegenerate :=
⟨fun h v hv => h v fun w => (M.toBilin'_apply' _ _).trans <| hv w, Matrix.Nondegenerate.toBilin'⟩
| Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 337 | 342 |
/-
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.Abelianization
import Mathlib.GroupTheory.Commutator.Finite
import Mathlib.GroupTheory.Transfer
import Mathlib.Algebra.Group.Pointwise.Fi... | calc
_ ≤ #(R * S) := Finset.card_image_le
_ ≤ #R * #S := Finset.card_mul_le
_ = H.index * S.card := congr_arg (· * S.card) ?_
calc
#R = Fintype.card R := (Fintype.card_coe R).symm
_ = _ := (Fintype.card_congr hR.rightQuotientEquiv).symm
_ = Fintype.card (G ⧸ H) := QuotientGroup.card_quotient... | Mathlib/GroupTheory/Schreier.lean | 139 | 158 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... |
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
| Mathlib/Topology/Order/Basic.lean | 351 | 355 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.RingTheory.Coprime.Lemmas
/-!
# Charac... | ⟨fun h =>
h.trans <|
Int.natCast_dvd_natCast.mp <|
(CharP.intCast_eq_zero_iff R (ringChar R) (Fintype.card R)).mp <|
mod_cast Nat.cast_card_eq_zero R,
fun h => ?_⟩
by_contra h₀
rcases exists_prime_addOrderOf_dvd_card p h with ⟨r, hr⟩
have hr₁ := addOrderOf_nsmul_eq_ze... | Mathlib/Algebra/CharP/CharAndCard.lean | 59 | 75 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice.Image
/-!
# Order connected components of a set
In this file we define `Set.ordConn... | rw [← not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent]
| Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 53 | 54 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
import Mathlib.CategoryTheory.Sites.Lo... | · change G.map _ (G.map _ _) = _
simp only [← FunctorToTypes.map_comp_apply, ← op_comp, Sigma.ι_desc]
@[deprecated (since := "2024-11-26")] alias regularTopology.isLocallySurjective_sheafOfTypes :=
regularTopology.isLocallySurjective_sheaf_of_types
lemma coherentTopology.presheafIsLocallySurjective_iff {F G... | Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean | 140 | 151 |
/-
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... | obtain ⟨C, hC⟩ := Asymptotics.isBigO_iff.mp h
obtain ⟨m, hm⟩ := eventually_atTop.mp hC
let C' := max C (max' (insert 0 (image (fun n : ℕ ↦ ‖f n‖ / (n : ℝ) ^ (x - 1)) (range m)))
(insert_nonempty 0 _))
| Mathlib/NumberTheory/LSeries/Basic.lean | 367 | 370 |
/-
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.Fintype.Card
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.OrderedM... | end LinearOrderedCancelCommMonoid
| Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 507 | 508 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | aesop (add norm Rel.inv_def)
| Mathlib/Data/Rel.lean | 344 | 344 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Logic.Equiv.Fin.... | exact this
open Cardinal Basis Submodule Function Set LinearMap
theorem Submodule.rank_sup_add_rank_inf_eq (s t : Submodule R M) :
Module.rank R (s ⊔ t : Submodule R M) + Module.rank R (s ⊓ t : Submodule R M) =
Module.rank R s + Module.rank R t := by
conv_rhs => enter [2]; rw [show t = (s ⊔ t) ⊓ t by simp... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 157 | 166 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | lemma toFinset_divisorsAntidiagonalList {n : ℕ} :
n.divisorsAntidiagonalList.toFinset = n.divisorsAntidiagonal := by
rw [divisorsAntidiagonalList, divisorsAntidiagonal, List.toFinset_filterMap (f_inj := by aesop),
List.toFinset_range'_1_1]
| Mathlib/NumberTheory/Divisors.lean | 134 | 137 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathl... | example :
(instTopologicalSpaceQuotient : TopologicalSpace <| M ⧸ S) =
(instSeminormedCommGroup S).toUniformSpace.toTopologicalSpace := rfl
| Mathlib/Analysis/Normed/Group/Quotient.lean | 218 | 220 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | | bit0 a, 1 => congr_arg bit0 (add_one a)
| Mathlib/Data/Num/Lemmas.lean | 84 | 84 |
/-
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.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... | theorem gcd_mono (h : s₁ ⊆ s₂) : s₂.gcd f ∣ s₁.gcd f :=
dvd_gcd fun _ hb ↦ gcd_dvd (h hb)
theorem gcd_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).gcd f = s.gcd (f ∘ g) := by
classical induction s using Finset.induction <;> simp [*]
| Mathlib/Algebra/GCDMonoid/Finset.lean | 166 | 171 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | rw [sub_eq_add_neg, sub_eq_add_neg, MultilinearMap.map_update_add, map_update_neg]
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,278 | 1,279 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 787 | 789 | |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Combinatorics.Additive.Frei... | section CancelCommMonoid
variable [CommMonoid α] [IsCancelMul α] {s : Set α} {a : α}
@[to_additive] lemma ThreeGPFree.eq_right (hs : ThreeGPFree s) :
∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → ∀ ⦃c⦄, c ∈ s → a * c = b * b → b = c := by
rintro a ha b hb c hc habc
obtain rfl := hs ha hb hc habc
simpa using habc.symm
| Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 155 | 164 |
/-
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.Data.Finsupp.Defs
/-!
# Locus of unequal values of finitely supported functions
Let `α N` be two Types, assume that `N` has a `0` and let `f g : α →₀ N... | @[simp]
theorem neLocus_zero_left : (0 : α →₀ N).neLocus f = f.support :=
| Mathlib/Data/Finsupp/NeLocus.lean | 69 | 70 |
/-
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 Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | simp only [v]
split_ifs with H
· rw [norm_zero]
| Mathlib/Analysis/SpecificLimits/Normed.lean | 566 | 568 |
/-
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.Analytic.CPolynomial
import Mathlib.Analysis.Analytic.Inverse
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.Deriv... | obtain rfl | hn := eq_or_ne n 0
· rw [zero_tsub]
refine HasFiniteFPowerSeriesOnBall.bound_zero_of_eq_zero (fun y hy ↦ ?_) h.r_pos fun n ↦ ?_
· rw [Filter.EventuallyEq.fderiv_eq (f := fun _ ↦ 0)]
· rw [fderiv_const, Pi.zero_apply]
| Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 498 | 502 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | (stdBasisMatrix i j a).map f = stdBasisMatrix i j (f a) := by
aesop (add unsafe unfold stdBasisMatrix)
end Zero
theorem stdBasisMatrix_add [AddZeroClass α] (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by
ext
simp only [stdBasisMatrix, of_apply]
... | Mathlib/Data/Matrix/Basis.lean | 65 | 79 |
/-
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.Calculus.Conformal.NormedSpace
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.... | -- let rot := c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ,
· refine Or.inl ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, ?_⟩
ext1
simp
· refine Or.inr ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, ?_⟩
ext1
simp
/-- A real continuous linear map on the complex plane is conformal if and only if the map or it... | Mathlib/Analysis/Complex/Conformal.lean | 97 | 115 |
/-
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... | namespace AddEquiv
| Mathlib/Data/Matrix/Basic.lean | 397 | 398 |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | obtain ⟨a, ha⟩ : (u \ s).Nonempty := by rw [← card_pos, card_sdiff hu₁]; omega
simp only [mem_sdiff] at ha
| Mathlib/Data/Finset/Card.lean | 567 | 568 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | map Sₘ (algebraMap R S)
(show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map) :=
Eq.symm <|
IsLocalization.map_unique _ (algebraMap Rₘ Sₘ) fun x => by
rw [← IsScalarTower.algebraMap_apply R S Sₘ, ← IsScalarTower.algebraMap_apply R Rₘ Sₘ]
| Mathlib/RingTheory/Localization/Basic.lean | 493 | 498 |
/-
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... | /-- A fiberwise inverse to `e`. This is the function `F → E b` that induces a local inverse
`B × F → TotalSpace F E` of `e` on `e.baseSet`. It is defined to be `0` outside `e.baseSet`. -/
protected noncomputable def symm (e : Pretrivialization F (π F E)) (b : B) (y : F) : E b :=
| Mathlib/Topology/FiberBundle/Trivialization.lean | 220 | 222 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
/-!
# Regular conditional probability distribution
We define the regular con... | open MeasureTheory
theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prodMk {Ω F} {mΩ : MeasurableSpace Ω}
(X : Ω → β) {μ : Measure Ω} [TopologicalSpace F] {f : Ω → F} (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) := by
refine ⟨fun x =>... | Mathlib/Probability/Kernel/CondDistrib.lean | 304 | 322 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
/-!
# Carathéodory's conv... | simp only [t.convexHull_eq, exists_prop, Set.mem_setOf_eq] at ht₃
obtain ⟨w, hw₁, hw₂, hw₃⟩ := ht₃
let t' := {i ∈ t | w i ≠ 0}
refine ⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩
· rw [Subtype.range_coe_subtype]
exact Subset.trans (Finset.filter_subset _ t) ht₁
· exact... | Mathlib/Analysis/Convex/Caratheodory.lean | 165 | 188 |
/-
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, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.... | | cons a s induct =>
rw [Multiset.prod_cons]
obtain ⟨k, rfl⟩ : a ∣ n := div a (Multiset.mem_cons_self a s)
apply mul_dvd_mul_left a
refine induct _ (fun a ha => h a (Multiset.mem_cons_of_mem ha)) (fun b b_in_s => ?_)
fun a => (Multiset.countP_le_of_le _ (Multiset.le_cons_self _ _)).trans (uniq a... | Mathlib/Algebra/BigOperators/Associated.lean | 124 | 130 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | rw [ofLE_mk_le_mk_of_comm (image.preComp h f)]
· infer_instance
· simp
| Mathlib/CategoryTheory/Subobject/Limits.lean | 343 | 346 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | instance isRotatedDecidable (l l' : List α) : Decidable (l ~r l') :=
decidable_of_iff' _ isRotated_iff_mem_map_range
instance {l l' : List α} : Decidable (IsRotated.setoid α l l') :=
| Mathlib/Data/List/Rotate.lean | 612 | 615 |
/-
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.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | mem_of_superset (inter_mem_inf hs ht) h
| Mathlib/Order/Filter/Basic.lean | 214 | 215 |
/-
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.ShortComplex.Homology
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# Homolo... | @[simp]
lemma cyclesMap'_sub :
cyclesMap' (φ - φ') h₁ h₂ = cyclesMap' φ h₁ h₂ -
cyclesMap' φ' h₁ h₂ := by
simp only [sub_eq_add_neg, cyclesMap'_add, cyclesMap'_neg]
| Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean | 132 | 136 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Al... | (structureSheaf R).1.obj (op U) → Localization.AtPrime x.asIdeal) =
fun s => s.1 ⟨x, hx⟩ :=
rfl
| Mathlib/AlgebraicGeometry/StructureSheaf.lean | 466 | 468 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 823 | 824 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... |
/-- The first coefficient of a composition of formal multilinear series is the composition of the
first coefficients seen as continuous linear maps. -/
theorem comp_coeff_one (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
(v : Fin 1 → E) : (q.comp p) 1 v = q 1 fun _i => p 1 v := by
hav... | Mathlib/Analysis/Analytic/Composition.lean | 248 | 255 |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.Group.PUnit
import Mathlib.Algebra.Group.ULift
import Mathlib.Analysis.Normed.Group.Basic
/-!
# Product of norme... | @[to_additive pi_nnnorm_const_le]
lemma pi_nnnorm_const_le' (a : E) : ‖fun _ : ι => a‖₊ ≤ ‖a‖₊ :=
pi_norm_const_le' _
@[to_additive (attr := simp) pi_norm_const]
lemma pi_norm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖ = ‖a‖ := by
simpa only [← dist_one_right] using dist_pi_const a 1
| Mathlib/Analysis/Normed/Group/Constructions.lean | 353 | 359 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | obtain rfl | hn := eq_or_ne n 0
· simp
refine (List.sorted_lt_range' _ _ Nat.one_ne_zero).filterMap ?_
simp only [Option.ite_none_right_eq_some, Option.some.injEq, gt_iff_lt, and_imp, Prod.forall,
Prod.mk.injEq]
| Mathlib/NumberTheory/Divisors.lean | 147 | 151 |
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