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 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 | 856 | 861 | |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Data.Set.Piecewise... |
@[to_additive (attr := simp)]
theorem const_eq_one : const ι a = 1 ↔ a = 1 :=
@const_inj _ _ _ _ 1
@[to_additive]
theorem const_ne_one : const ι a ≠ 1 ↔ a ≠ 1 :=
Iff.not const_eq_one
end Function
section Piecewise
@[to_additive]
theorem Set.piecewise_mul [∀ i, Mul (f i)] (s : Set I) [∀ i, Decidable (i ∈ s)]
... | Mathlib/Algebra/Group/Pi/Lemmas.lean | 371 | 401 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yaël Dillies
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Data.Set.MulAntidiagonal
import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-! # Multiplicat... | obtain rfl :=
(hs.min_le hns has).eq_of_not_lt fun hlt =>
(mul_lt_mul_of_lt_of_le hlt <| ht.min_le hnt hat).ne' hst
exact ⟨rfl, mul_left_cancel hst⟩
| Mathlib/Data/Finset/MulAntidiagonal.lean | 100 | 103 |
/-
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.CategoryTheory.Shift.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-! Sequences of functors from a category equipped with a shift
Let `F : C... | simp [F.shiftIso_zero a]
lemma shiftIso_add (n m a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') :
F.shiftIso (m + n) a a'' (by rw [add_assoc, ha', ha'']) =
isoWhiskerRight (shiftFunctorAdd C m n) _ ≪≫ Functor.associator _ _ _ ≪≫
| Mathlib/CategoryTheory/Shift/ShiftSequence.lean | 136 | 140 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Analysis.Normed.Group.Constructions
import Mathlib.Analysis.Normed.Group.Rat
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topolo... |
theorem norm_sup_le_add (x y : α) : ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖ := by
have h : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_sup_sub_sup_le_add_norm x y 0 0
simpa only [sup_idem, sub_zero] using h
-- see Note [lower instance priority]
/-- Let `α` be a normed lattice ordered group. Then the infimum is jointly continuous.
-/... | Mathlib/Analysis/Normed/Order/Lattice.lean | 134 | 145 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | apply lt_of_le_of_lt _ hn
rw [norm_le_pow_iff_mem_span_pow]
apply appr_spec
| Mathlib/NumberTheory/Padics/RingHoms.lean | 444 | 447 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... |
theorem setToL1S_mono (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
(h_add : FinMeasAdditive μ T)
(hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁ₛ[μ] G''}
(hfg : f ≤ g) : setToL1S T f ≤ setToL1S T g := by
rw [← sub_nonneg] at hfg ⊢
rw [← setToL1S_sub h_zero h_add... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 251 | 263 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... | Mathlib/Order/Irreducible.lean | 107 | 107 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | theorem inverse_counitInv_comp (e : C ≌ D) (Y : D) :
e.inverse.map (e.counitInv.app Y) ≫ e.unitInv.app (e.inverse.obj Y) = 𝟙 (e.inverse.obj Y) := by
simpa using Iso.inv_eq_inv
(e.unitIso.app (e.inverse.obj Y) ≪≫ e.inverse.mapIso (e.counitIso.app Y)) (Iso.refl _)
| Mathlib/CategoryTheory/Equivalence.lean | 214 | 217 |
/-
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... | Mathlib/Data/Num/Lemmas.lean | 1,531 | 1,531 | |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | intro i
calc
↑p ^ 2 ∣ (↑p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]
_ ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) := by
simp only [sq_dvd_add_pow_sub_sub (↑p * b) a i, ← sub_sub]
simp_rw [← mem_span_singleton, ← Ideal.Quotient.eq] at *
let s : R := (p : R)^2
calc... | Mathlib/NumberTheory/Multiplicity.lean | 82 | 146 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Normed.Module.FiniteDimension
/-!
# Infinitely smooth "bump" func... |
theorem _root_.ContDiff.contDiffBump {c g : X → E} {f : ∀ x, ContDiffBump (c x)}
| Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 191 | 192 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | (by rintro rfl; exact .head _) (by rintro rfl; exact .head _)
else
(mem_or_mem_of_zipWith_swap_prod_ne h).imp (.tail _) (.tail _)
theorem zipWith_swap_prod_support' (l l' : List α) :
{ x | (zipWith swap l l').prod x ≠ x } ≤ l.toFinset ⊔ l'.toFinset := fun _ h ↦ by
simpa using mem_or_mem_of_zipWi... | Mathlib/GroupTheory/Perm/List.lean | 72 | 81 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | ⟨q, H, fun _ H2 ↦ ExpChar.eq H2 H⟩
instance ringExpChar.expChar [Ring R] [IsDomain R] : ExpChar R (ringExpChar R) := by
obtain ⟨q, _⟩ := ExpChar.exists R
rwa [ringExpChar.eq R q]
variable {R} in
lemma ringExpChar.of_eq [Ring R] [IsDomain R] {q : ℕ} (h : ringExpChar R = q) : ExpChar R q :=
h ▸ ringExpChar.expC... | Mathlib/Algebra/CharP/Defs.lean | 415 | 423 |
/-
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... | rw [IsRoot, this, eval_C] at h
simp only [h, RingHom.map_zero] at this
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 72 | 73 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | Mathlib/Data/Nat/PartENat.lean | 795 | 796 | |
/-
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... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,259 | 1,260 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | exact ↑(map fun _ => (1 : ℤ)))
invFun x := of () ^ x
left_inv := by
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 829 | 831 |
/-
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.Data.Finset.Lattice.Fold
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
/-!
# Symmetric powers of a finset
This file defines the sym... | simp only [mem_product, h, and_self, true_and]
· rintro ⟨⟨a, b⟩, h⟩
simp only [mem_product, Sym2.eq_iff] at h
obtain ⟨h, (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)⟩ := h
| Mathlib/Data/Finset/Sym.lean | 139 | 142 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Equalizers and coequalizers
This file defines (co)equalizers a... | assoc := assoc
@[simp]
theorem walkingParallelPairHom_id (X : WalkingParallelPair) : WalkingParallelPairHom.id X = 𝟙 X :=
| Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 105 | 108 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | simp [mul_apply, hf, g.injective hg])
fun hg =>
(h (f x)).elim (fun hf => by simp [mul_apply, f.injective hf, hg]) fun hf => by
simp [mul_apply, hf, hg]
@[simp]
theorem disjoint_one_left (f : Perm α) : Disjoint 1 f := fun _ => Or.inl rfl
@[simp]
| Mathlib/GroupTheory/Perm/Support.lean | 63 | 71 |
/-
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.Algebra.Group.Pi.Basic
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... |
@[simp]
| Mathlib/Order/Interval/Set/Pi.lean | 38 | 39 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... | rw [epi_iff_surjective_up_to_refinements]
intro A g₁
obtain ⟨A₁, π₁, _, f₂, h₁⟩ :=
surjective_up_to_refinements_of_epi (app' φ 2 _) (g₁ ≫ R₂.map' 1 2)
have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by
rw [← cancel_mono (app' φ 3 _), assoc, zero_comp, NatTrans.naturality, ← reassoc_of% h₁,
← R₂.map'_comp 1 2 3,... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 95 | 120 |
/-
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 | 3,149 | 3,153 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | section
| Mathlib/Topology/UniformSpace/Basic.lean | 548 | 549 |
/-
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.Field.Equiv
import Mathlib.Algebra.Field.Subfield.Basic
import Mathlib.Algebra.Order.Ring... |
theorem mk'_mk_eq_div {r s} (hs : s ∈ nonZeroDivisors A) :
mk' K r ⟨s, hs⟩ = algebraMap A K r / algebraMap A K s :=
haveI := (algebraMap A K).domain_nontrivial
| Mathlib/RingTheory/Localization/FractionRing.lean | 167 | 170 |
/-
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... | · rcases h with ⟨d, hd⟩
rcases gcd_dvd_right a b with ⟨e, he⟩
| Mathlib/Algebra/GCDMonoid/Basic.lean | 879 | 880 |
/-
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.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | theorem essSup_mono_measure' {α : Type*} {β : Type*} {_ : MeasurableSpace α}
{μ ν : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} (hμν : ν ≤ μ) :
essSup f ν ≤ essSup f μ :=
| Mathlib/MeasureTheory/Function/EssSup.lean | 200 | 202 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Pythagorean Triples
Th... | (To be applied in the case where `K = ℚ`.) -/
def circleEquivGen (hk : ∀ x : K, 1 + x ^ 2 ≠ 0) :
K ≃ { p : K × K // p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } where
toFun x :=
⟨⟨2 * x / (1 + x ^ 2), (1 - x ^ 2) / (1 + x ^ 2)⟩, by
field_simp [hk x, div_pow]
ring, by
simp only [Ne, div_eq_iff (hk x),... | Mathlib/NumberTheory/PythagoreanTriples.lean | 255 | 263 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... |
theorem models_formula_iff {φ : L.Formula α} :
T ⊨ᵇ φ ↔ ∀ (M : ModelType.{u, v, max u v w} T) (v : α → M), φ.Realize v :=
forall_congr' fun _ => forall_congr' fun _ => Unique.forall_iff
theorem models_sentence_iff {φ : L.Sentence} : T ⊨ᵇ φ ↔ ∀ M : ModelType.{u, v, max u v} T, M ⊨ φ :=
models_formula_iff.trans... | Mathlib/ModelTheory/Satisfiability.lean | 287 | 295 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Ba... | lemma poissonPMFReal_pos {r : ℝ≥0} {n : ℕ} (hr : 0 < r) : 0 < poissonPMFReal r n := by
rw [poissonPMFReal]
positivity
| Mathlib/Probability/Distributions/Poisson.lean | 49 | 52 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
/-!
# Exterior Algebras
We construct the e... |
variable {R M}
/-- The canonical image of the `TensorAlgebra` in the `ExteriorAlgebra`, which maps
`TensorAlgebra.ι R x` to `ExteriorAlgebra.ι R x`. -/
def toExterior : TensorAlgebra R M →ₐ[R] ExteriorAlgebra R M :=
TensorAlgebra.lift R (ExteriorAlgebra.ι R : M →ₗ[R] ExteriorAlgebra R M)
| Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 452 | 459 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient.Basic
/-!
# Isomorphism theorems for modules.
* The Noether's first, second, a... | Mathlib/LinearAlgebra/Isomorphisms.lean | 191 | 195 | |
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | rw [← eqv.left_inv π]
intro f₁ f₂
let eqv' := homEquivAmalgamation (eqv π).2
apply eqv'.injective
ext
apply h _ (eqv π).1 <;> exact (eqv' _).2
/-- A presheaf `P` is a sheaf for the Grothendieck topology `J` iff for every covering sieve
`S` of `J`, the natural cone associated to `P` and `S` ... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 170 | 189 |
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.LinearAlgebra.Multilinear.Basis
/-!
# Alte... | @[simp]
theorem domLCongr_symm (e : M ≃ₗ[R] M₂) : (domLCongr R N ι S e).symm = domLCongr R N ι S e.symm :=
rfl
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 564 | 567 |
/-
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.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | simp [vadd_vsub_assoc, vsub_vadd_eq_vsub_sub, add_comm]
lemma weightedVSubOfPoint_smul {G : Type*} [Group G] [DistribMulAction G V] [SMulCommClass G k V]
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 79 | 81 |
/-
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.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... |
section
variable {X Y : C} [HasBinaryBiproduct X Y]
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 421 | 424 |
/-
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.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | protected def sigma (s : Multiset α) (t : ∀ a, Multiset (σ a)) : Multiset (Σa, σ a) :=
s.bind fun a => (t a).map <| Sigma.mk a
| Mathlib/Data/Multiset/Bind.lean | 306 | 307 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | exact derivWithin_mul_const_field u
theorem deriv_const_mul (c : 𝔸) (hd : DifferentiableAt 𝕜 d x) :
deriv (fun y => c * d y) x = c * deriv d x :=
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 301 | 304 |
/-
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.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.RingTheory.MatrixPolynomialAlgebra
/-!
# Characteristic polynomials... | charmatrix (M.map f) = (charmatrix M).map (Polynomial.map f) := by
ext i j
by_cases h : i = j <;> simp [h, charmatrix, diagonal]
lemma charmatrix_fromBlocks :
| Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 79 | 83 |
/-
Copyright (c) 2021 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.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois.Basic
/-!
# Cyclotomic extens... | refine ⟨fun hn => ((isCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => ?_⟩
replace h := ((isCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in ... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 174 | 188 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.FieldTheory.Perfect
/-!
# The perfect closure of a characteristic `p` ring
## Main definitions
- `Perfec... | qsmul_def := fun _ _ => rfl
instance instField : Field (PerfectClosure K p) :=
{ (inferInstance : DivisionRing (PerfectClosure K p)),
(inferInstance : CommRing (PerfectClosure K p)) with }
| Mathlib/FieldTheory/PerfectClosure.lean | 485 | 489 |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subring.Defs
import Mathlib.Algebra.Ring.Subsemiring.Bas... |
theorem comap_inf (s t : Subring S) (f : R →+* S) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
| Mathlib/Algebra/Ring/Subring/Basic.lean | 679 | 680 |
/-
Copyright (c) 2024 Lean FRO LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.Catego... | rw [← op_inv_associator, ← op_whiskerRight, ← op_comp, ← op_whiskerLeft, ← op_comp,
comul_assoc_flip, op_comp, op_comp_assoc]
rfl
| Mathlib/CategoryTheory/Monoidal/Comon_.lean | 261 | 264 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,747 | 1,754 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | theorem fderiv_comp_deriv_of_eq (hl : DifferentiableAt 𝕜 l y) (hf : DifferentiableAt 𝕜 f x)
(hy : y = f x) :
deriv (l ∘ f) x = (fderiv 𝕜 l (f x) : F → E) (deriv f x) := by
rw [hy] at hl; exact fderiv_comp_deriv x hl hf
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 404 | 408 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 3,130 | 3,132 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Module.Convex
import Mathlib.Analysis.Normed.Module.Ray
import Mathlib.Analysis.NormedSpace.Pointwise
/-!
# Strictly conv... | rw [← norm_pos_iff] at hx hy
have hxy : 0 < ‖x‖ + ‖y‖ := add_pos hx hy
have :=
combo_mem_ball_of_ne (inv_norm_smul_mem_unitClosedBall x)
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 170 | 173 |
/-
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... | end WeaklyRegular
namespace Regular
variable [TopologicalSpace α]
instance zero : Regular (0 : Measure α) :=
⟨fun _ _ _r hr => ⟨∅, empty_subset _, isCompact_empty, hr⟩⟩
| Mathlib/MeasureTheory/Measure/Regular.lean | 973 | 980 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... | have h₀ : (ℙ.map X).prod (ℙ.map Y) =
(μ.prod ν).withDensity fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2 :=
prod_eq fun s t hs ht ↦ by rw [withDensity_apply _ (hs.prod ht), ← prod_restrict,
lintegral_prod_mul (measurable_pdf X ℙ μ).aemeasurable (measurable_pdf Y ℙ ν).aemeasurable,
| Mathlib/Probability/Density.lean | 311 | 314 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Fiber bundles
Mathemat... |
@[simp, mfld_simps]
theorem mem_localTrivAt_baseSet (b : B) : b ∈ (Z.localTrivAt b).baseSet := by
| Mathlib/Topology/FiberBundle/Basic.lean | 668 | 670 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
import Mathlib.MeasureTh... | theorem lintegral_rpow_funMulInvSnorm_eq_one {p : ℝ} (hp0_lt : 0 < p) {f : α → ℝ≥0∞}
(hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) :
∫⁻ c, funMulInvSnorm f p μ c ^ p ∂μ = 1 := by
simp_rw [funMulInvSnorm_rpow hp0_lt]
rw [lintegral_mul_const', ENNReal.mul_inv_cancel hf_nonzero hf_top... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 93 | 98 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | [∀ i, Fintype (ι i)] {𝕜 : Type*} [RCLike 𝕜] {E : η → Type*} [∀ i, NormedAddCommGroup (E i)]
[∀ i, InnerProductSpace 𝕜 (E i)] (B : ∀ i, OrthonormalBasis (ι i) 𝕜 (E i))
(j : (i : η) × (ι i)) :
Pi.orthonormalBasis B j = (WithLp.equiv _ _).symm (Pi.single _ (B j.fst j.snd)) := by
classical
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 540 | 544 |
/-
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.Topology.MetricSpace.Pseudo.Lemmas
import Mathlib.Topology.EMetricSpace.Basic
... | · exact lt_of_lt_of_le (hN _ h) (le_add_of_nonneg_left R.2)
· have : _ ≤ R := Finset.le_sup (Finset.mem_range.2 h)
exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one)
/-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient. -/
theorem cauchySeq_if... | Mathlib/Topology/MetricSpace/Cauchy.lean | 113 | 123 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Trace.Basic
import Mathlib.RingTheory.Norm.Basic
/-!
# Discriminant of a famil... | rw [Basis.equivFun_apply] at hr
rw [← smul_assoc, ← hr, algebraMap_smul]
refine Subalgebra.smul_mem _ ?_ _
rw [B.basis_eq_pow i]
exact Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton _)) _
intro i
rw [← H, ← mulVec_smul] at cramer
replace cramer := congr_arg (mulVec (traceMatrix K B.bas... | Mathlib/RingTheory/Discriminant.lean | 277 | 311 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.NumberTheory.LSeries.MellinEqDirichlet
i... | simpa using oddKernel_neg 0 x
lemma sinKernel_neg (a : UnitAddCircle) (x : ℝ) :
sinKernel (-a) x = -sinKernel a x := by
| Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 143 | 146 |
/-
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.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... |
/-- If `S / R` and `S' / R'` are algebras, `i : R →+* R'` is a surjective ring homomorphism,
`j : S →+* S'` is an injective ring homomorphism, such that `R → R' → S'` and `R → S → S'` commute,
then the rank of `S / R` is smaller than or equal to the rank of `S' / R'`. -/
theorem lift_rank_le_of_surjective_injective
| Mathlib/LinearAlgebra/Dimension/Basic.lean | 221 | 225 |
/-
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.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | rw [← toFinsupp_inj, toFinsupp_monomial, toFinsupp_monomial, Finsupp.single_eq_single_iff]
| Mathlib/Algebra/Polynomial/Basic.lean | 443 | 444 |
/-
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... | | pos p, 0 => show pos (p + 1) = succ (pos p + 0) by rw [PosNum.add_one, add_zero, succ, succ']
| pos _, pos _ => congr_arg pos (PosNum.add_succ _ _)
theorem bit0_of_bit0 : ∀ n : Num, n + n = n.bit0
| 0 => rfl
| pos p => congr_arg pos p.bit0_of_bit0
| Mathlib/Data/Num/Lemmas.lean | 189 | 194 |
/-
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.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... | (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y :=
hs.eq hx hy fun H => h <| H ha hb hab
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
| Mathlib/Analysis/Convex/Strict.lean | 67 | 70 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | Mathlib/Data/Bool/Basic.lean | 224 | 224 | |
/-
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... | @[simp]
theorem add_tsub_cancel_left (a b : α) : a + b - a = b :=
| Mathlib/Algebra/Order/Sub/Defs.lean | 330 | 331 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | theorem maximal_mem_image_antitone (hf : ∀ ⦃x y⦄, x ∈ s → y ∈ s → (f x ≤ f y ↔ y ≤ x))
(hx : Maximal (· ∈ s) x) : Minimal (· ∈ f '' s) (f x) :=
maximal_mem_image_monotone (β := βᵒᵈ) (fun _ _ h h' ↦ hf h' h) hx
theorem minimal_mem_image_antitone_iff (ha : a ∈ s)
| Mathlib/Order/Minimal.lean | 454 | 458 |
/-
Copyright (c) 2023 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.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | nonrec lemma is3Clique_iff [NoAccidental t] {s : Finset (α ⊕ β ⊕ γ)} :
(graph t).IsNClique 3 s ↔ ∃ x, x ∈ t ∧ toTriangle x = s := by
refine ⟨fun h ↦ ?_, ?_⟩
· rw [is3Clique_iff] at h
obtain ⟨x, y, z, hxy, hxz, hyz, rfl⟩ := h
obtain ⟨a, b, c, habc, hab, hac, hbc⟩ := graph_triple hxy hxz hyz
refine ⟨(... | Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 167 | 179 |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Math... | Mathlib/Combinatorics/Quiver/Covering.lean | 312 | 314 | |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | simp [eLpNormEssSup]
@[simp]
theorem eLpNorm_measure_zero {f : α → ε} : eLpNorm f p (0 : Measure α) = 0 := by
by_cases h0 : p = 0
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 152 | 156 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | rw [h]
@[simp]
theorem coe_ofRepr [DecidableEq ι] (e : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι) :
⇑(OrthonormalBasis.ofRepr e) = fun i => e.symm (EuclideanSpace.single i (1 : 𝕜)) := by
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 350 | 354 |
/-
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... | · exact ⟨fun _ _ ↦ hret, fun _ _ ↦ hgo⟩
· exact ⟨fun _ _ ↦ hret, fun _ _ ↦ hgo⟩
· exact ⟨⟨fun _ _ ↦ hret, fun _ _ ↦ hret⟩, fun _ _ ↦ hgo⟩
| Mathlib/Computability/TuringMachine.lean | 769 | 771 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... |
protected theorem zero_ne_one : 0 ≠ (1 : ℚ) := by
| Mathlib/Data/Rat/Defs.lean | 268 | 269 |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered systems
This file de... | haveI : ∀ j, Nonempty (F'.obj j) := hne
obtain ⟨⟨u, hu⟩⟩ := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} F'
exact ⟨u, hu⟩
/-- The cofiltered limit of nonempty finite types is nonempty.
See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_s... | Mathlib/CategoryTheory/CofilteredSystem.lean | 68 | 76 |
/-
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.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | theorem f_add_nat_ge (hf_conv : ConvexOn ℝ (Ioi 0) f)
(hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : 2 ≤ n) (hx : 0 < x) :
f n + x * log (n - 1) ≤ f (n + x) := by
have npos : 0 < (n : ℝ) - 1 := by rw [← Nat.cast_one, sub_pos, Nat.cast_lt]; omega
have c :=
(convexOn_iff_slope_mono_adjacent.m... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 176 | 184 |
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | theorem repr_sub (a b) : repr (a - b) = repr a - repr b :=
| Mathlib/SetTheory/Ordinal/Notation.lean | 1,234 | 1,234 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
/-! # LDL decomposition
This file proves the LDL-decom... | star_dotProduct_star, star_star]
rfl
/-- The lower triangular matrix `L` of the LDL decomposition. -/
noncomputable def LDL.lower :=
(LDL.lowerInv hS)⁻¹
/-- **LDL decomposition**: any positive definite matrix `S` can be
decomposed as `S = LDLᴴ` where `L` is a lower-triangular matrix and `D` is a diagonal ... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 102 | 113 |
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
import Mathlib.MeasureTheory.Function.LpSpace.Complete
import Mathlib.MeasureThe... | suffices Tendsto (fun n => ∫⁻ x, ‖approxOn f hf s y₀ h₀ n x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0) by
simp only [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp_ne_top]
convert continuous_rpow_const.continuousAt.tendsto.comp this
simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]
-- We simply check the conditions ... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 93 | 135 |
/-
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.Relation
/-!
# Parallel computation
Parallel computation of a computable sequence of computations by
a diagonal enumeration.
The imp... | intro l
induction' l with c l IH <;> simp only [parallel.aux2, List.foldr]
· intro a h
rcases h with ⟨c, hn, _⟩
exact False.elim <| List.not_mem_nil hn
· simp only [parallel.aux2] at IH
-- Porting note: `revert IH` & `intro IH` are required.
revert IH
cases List.foldr (fun ... | Mathlib/Data/Seq/Parallel.lean | 189 | 266 |
/-
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 | 213 | 219 | |
/-
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.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | exact NNReal.inner_le_weight_mul_Lp _ hp _ _
/-- **Weighted Hölder inequality** in terms of `Finset.expect`. -/
lemma compact_inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) {w f : ι → ℝ}
| Mathlib/Analysis/MeanInequalities.lean | 756 | 759 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Defs
/-!
# Locus of unequal values of finitely supported dependent functions
Let `N : α → Type*` be a type family, assume that `N ... |
section NeLocusAndMaps
| Mathlib/Data/DFinsupp/NeLocus.lean | 72 | 74 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | [RingHomClass F R S] {x : R} (hx : x ∈ nthRootsFinset n 1) (f : F) :
f x ∈ nthRootsFinset n 1 := by
rw [← (map_one f)]
| Mathlib/Algebra/Polynomial/Roots.lean | 342 | 344 |
/-
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.Abelian.Basic
/-!
# Abelian categories have homology
In this file, it is shown that all s... | lemma kernel_ι_comp_cokernel_π_comp_cokernelToAbelianCoimage :
(kernel.ι S.g ≫ cokernel.π S.f) ≫ S.cokernelToAbelianCoimage = 0 := by simp
| Mathlib/Algebra/Homology/ShortComplex/Abelian.lean | 123 | 124 |
/-
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 | 406 | 407 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) :
Icc a₁ b₁ ⊂ Icc a₂ b₂ := by
rw [← coe_ssubset, coe_Icc, coe_Icc]
| Mathlib/Order/Interval/Finset/Basic.lean | 230 | 232 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.MorphismProperty.Concrete
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Lim... | infer_instance
variable [HasStrongEpiMonoFactorisations C] [(forget C).PreservesMonomorphisms]
[(forget C).PreservesEpimorphisms]
| Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean | 112 | 115 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.GroupTheory.Subsemigroup.Center
import Mathlib.RingTheory... | [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
open NonUnitalRingHom
@[simp]
theorem range_subtype (s : NonUnitalSubring R) : (NonUnitalSubringClass.subtype s).range = s :=
SetLike.coe_injective <| (coe_srange _).trans Subtype.range_coe
theorem range_fst : NonUnitalRingHom.srange (fst R S) = ⊤ :=
NonUnita... | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | 719 | 744 |
/-
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... | target_eq := by simp [target_eq, prod_univ]
proj_toFun p hp := e.proj_toFun p hp.1
section Piecewise
theorem frontier_preimage (e : Trivialization F proj) (s : Set B) :
e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s) := by
rw [← (e.isImage_preimage_prod s).frontier.preimage_eq, frontier... | Mathlib/Topology/FiberBundle/Trivialization.lean | 657 | 665 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | exact hl.1
· intro _ e
obtain ⟨a, ha, rfl⟩ := mem_map.mp e
exact Set.mem_image_of_mem _ (hl.2 _ ha)
· rw [length_map]
variable (s)
@[simp]
theorem chainHeight_dual : (ofDual ⁻¹' s).chainHeight = s.chainHeight := by
apply le_antisymm <;>
· rw [chainHeight_le_chainHeight_iff]
rintro l ... | Mathlib/Order/Height.lean | 212 | 237 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | /-- The convex body equal to the set of points `x : mixedSpace K` such that
`∑ w real, ‖x w‖ + 2 * ∑ w complex, ‖x w‖ ≤ B`. -/
abbrev convexBodySum : Set (mixedSpace K) := { x | convexBodySumFun x ≤ B }
open scoped Classical in
theorem convexBodySum_volume_eq_zero_of_le_zero {B} (hB : B ≤ 0) :
volume (convexBod... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 316 | 324 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.Topology.Algebr... | lemma valuation_eq_intValuationDef (r : R) : v.valuation K r = v.intValuationDef r :=
Valuation.extendToLocalization_apply_map_apply ..
| Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 285 | 286 |
/-
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... | continuous_toFun := continuous_id.quotient_map' <| by
rw [inseparableSetoid_ring]; exact fun _ _ ↦ id
continuous_invFun := continuous_id.quotient_map' <| by
rw [inseparableSetoid_ring]; exact fun _ _ ↦ id
| Mathlib/Topology/Algebra/UniformRing.lean | 242 | 245 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | simp only [SizeOf.sizeOf, _sizeOf_1]
induction' xs with x xs
· simp [dedupKeys]
· simp only [dedupKeys_cons, kinsert_def, Nat.add_le_add_iff_left, Sigma.eta]
trans
· apply sizeOf_kerase
· assumption
| Mathlib/Data/List/Sigma.lean | 619 | 626 |
/-
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.Basic
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
/-!
# The fold operation for a commutative associative operation ... |
theorem fold_hom {op' : γ → γ → γ} [Std.Commutative op'] [Std.Associative op'] {m : β → γ}
(hm : ∀ x y, m (op x y) = op' (m x) (m y)) :
| Mathlib/Data/Finset/Fold.lean | 83 | 85 |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | ⟨fun hn _ hm ↦ isRoot_iterate_derivative_of_lt_rootMultiplicity <| Nat.lt_of_le_of_lt hm hn,
fun hr ↦ lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors' h hr hnzd⟩
theorem one_lt_rootMultiplicity_iff_isRoot_iterate_derivative
{p : R[X]} {t : R} (h : p ≠ 0) :
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 118 | 122 |
/-
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... |
theorem T_neg_two : T R (-2) = 2 * X ^ 2 - 1 := by simp [T_two]
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 131 | 132 |
/-
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.HomotopyCategory.MappingCone
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift
import Mathlib.CategoryTheory.Triangulated.Functor... | open Preadditive
variable (G : C ⥤ D) [G.Additive]
| Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean | 388 | 391 |
/-
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.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | eq_of_veq <| Multiset.map_cons f a s.val
@[simp]
| Mathlib/Data/Finset/Image.lean | 235 | 237 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | theorem comp_right_hasFDerivAt_iff {f : F → G} {x : E} {f' : F →L[𝕜] G} :
HasFDerivAt (f ∘ iso) (f'.comp (iso : E →L[𝕜] F)) x ↔ HasFDerivAt f f' (iso x) := by
simp only [← hasFDerivWithinAt_univ, ← comp_right_hasFDerivWithinAt_iff, preimage_univ]
| Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 202 | 205 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Antoine Chambert-Loir
-/
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.LinearAlgebra.Finsupp.SumProd
/-!... | finsuppScalarRight R M ι (m ⊗ₜ[R] p) i = p i • m := by
simp [finsuppScalarRight]
lemma finsuppScalarRight_apply_tmul (m : M) (p : ι →₀ R) :
| Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 217 | 220 |
/-
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.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | theorem one_half_pos : (0 : α) < 1 / 2 :=
half_pos zero_lt_one
| Mathlib/Algebra/Order/Field/Basic.lean | 146 | 148 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | noncomputable instance uniqueOneToPGameLeftMoves : Unique (toPGame 1).LeftMoves :=
(Equiv.cast <| toPGame_leftMoves 1).unique
@[simp]
theorem one_toPGame_leftMoves_default_eq :
| Mathlib/SetTheory/Game/Ordinal.lean | 83 | 87 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | def mapAlgHom (φ : K[X] →ₐ[S] R[X]) (hφ : K[X]⁰ ≤ R[X]⁰.comap φ) : RatFunc K →ₐ[S] RatFunc R :=
{ mapRingHom φ hφ with
commutes' := fun r => by
simp_rw [RingHom.toFun_eq_coe, coe_mapRingHom_eq_coe_map, algebraMap_apply r, map_apply_div,
map_one, AlgHom.commutes] }
| Mathlib/FieldTheory/RatFunc/Basic.lean | 612 | 616 |
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