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 |
|---|---|---|---|---|
/-
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.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | private theorem gauge_set_bddBelow : BddBelow { r : ℝ | 0 < r ∧ x ∈ r • s } :=
⟨0, fun _ hr => hr.1.le⟩
| Mathlib/Analysis/Convex/Gauge.lean | 66 | 68 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
/-!
# Towers of algebras
We set up the basic theory of algebra towers.
An algebra tower A/S/R is... |
section Ring
variable {R S}
variable [CommRing R] [Ring S] [Algebra R S]
-- Porting note: Needed to add Algebra.toModule below
theorem Basis.algebraMap_injective {ι : Type*} [NoZeroDivisors R] [Nontrivial S]
(b : @Basis ι R S _ _ Algebra.toModule) : Function.Injective (algebraMap R S) :=
have : NoZeroSMulDivis... | Mathlib/RingTheory/AlgebraTower.lean | 160 | 170 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Injective.Resolution
/-!
# Right-derived functors
We define the ... | simp only [← cancel_mono (HomologicalComplex.iCycles _ _),
Functor.mapHomologicalComplex_obj_X, assoc, toRightDerivedZero'_comp_iCycles,
CochainComplex.single₀_obj_zero, HomologicalComplex.cyclesMap_i,
Functor.mapHomologicalComplex_map_f, toRightDerivedZero'_comp_iCycles_assoc,
← F.map_comp, comm]
in... | Mathlib/CategoryTheory/Abelian/RightDerived.lean | 269 | 281 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.LiftingProperties.Basic
import Mathlib.CategoryTheory.Adjunction.Basic
/-!
# Lifting properties and adjunction
In this file, we obtain `Adjunct... | · rw [← sq.left_adjoint_hasLift_iff adj]
infer_instance
· rw [← sq.right_adjoint_hasLift_iff adj]
infer_instance
end Adjunction
| Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean | 126 | 132 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... | ext fun x ↦ by simp [not_or]
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 106 | 107 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 307 | 309 | |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.NumberTheory.NumberField.Discriminant.Defs
/-!
# ... | the formula uses `1 / 2 = 0` and `0 - 1 = 0`. It is useful only to have a uniform result.
See also `IsCyclotomicExtension.discr_prime_pow_eq_unit_mul_pow`. -/
theorem discr_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} K L] [hp : Fact (p : ℕ).Prime]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) (hirr : Irreducible (cyclotom... | Mathlib/NumberTheory/Cyclotomic/Discriminant.lean | 128 | 132 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... | theorem UniqueDiffWithinAt.accPt [Nontrivial E] (h : UniqueDiffWithinAt 𝕜 s x) : AccPt x (𝓟 s) := by
by_contra! h'
have : Dense (Submodule.span 𝕜 (0 : Set E) : Set E) :=
h.1.mono <| by gcongr; exact tangentConeAt_subset_zero h'
simp [dense_iff_closure_eq] at this
/-- In a proper space, the tangent cone at... | Mathlib/Analysis/Calculus/TangentCone.lean | 310 | 318 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | @[simp]
theorem symmDiff_hnot_self : a ∆ (￢a) = ⊤ := by rw [symmDiff_comm, hnot_symmDiff_self]
| Mathlib/Order/SymmDiff.lean | 297 | 299 |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | · rw [(ext rfl hi : x = x.re)]
rcases lt_trichotomy x.re 0 with (hr | hr | hr)
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 439 | 440 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | theorem isVonNBounded_iff {s : Set E} : Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s := by
refine ⟨fun h ↦ ?_, isVonNBounded_of_isBounded _⟩
rcases (h (Metric.ball_mem_nhds 0 zero_lt_one)).exists_pos with ⟨ρ, hρ, hρball⟩
rcases NormedField.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩
specialize hρball a ha.le
rw ... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 460 | 469 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Finite G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ Nat.card G) (H : Subgroup G... | Mathlib/GroupTheory/Sylow.lean | 630 | 645 |
/-
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... | simp only [isLittleO_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_atTop_iff, false_imp_iff,
imp_true_iff, tendsto_exp_comp_nhds_zero, neg_sub]
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 396 | 398 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Data.SetLike.Basic
/-!
# Order intervals
This file defines (nonempty) close... | else ⊥
sInf_le := fun s₁ s ha => by
| Mathlib/Order/Interval/Basic.lean | 623 | 624 |
/-
Copyright (c) 2022 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.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Order.Field
impo... | end SimpleGraph
/-! ### Uniform partitions -/
variable [DecidableEq α] {A : Finset α} (P : Finpartition A) (G : SimpleGraph α)
| Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 182 | 187 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.DirectSum.Internal
... | (decompose 𝒜 (a * b) n : A) = a * decompose 𝒜 b (n - i) := by
lift a to 𝒜 i using a_mem
rwa [decompose_mul, decompose_coe, coe_of_mul_apply_of_le]
| Mathlib/RingTheory/GradedAlgebra/Basic.lean | 326 | 329 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | theorem deriv_sqrt_mul_log (x : ℝ) :
deriv (fun x => √x * log x) x = (2 + log x) / (2 * √x) := by
rcases lt_or_le 0 x with hx | hx
· exact (hasDerivAt_sqrt_mul_log hx.ne').deriv
· rw [sqrt_eq_zero_of_nonpos hx, mul_zero, div_zero]
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 115 | 119 |
/-
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.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | unfold span
| Mathlib/RingTheory/Ideal/Operations.lean | 428 | 428 |
/-
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... | theorem Topology.IsInducing.codRestrict {e : X → Y} (he : IsInducing e) {s : Set Y}
(hs : ∀ x, e x ∈ s) : IsInducing (codRestrict e s hs) :=
he.of_comp (he.continuous.codRestrict hs) continuous_subtype_val
@[deprecated (since := "2024-10-28")] alias Inducing.codRestrict := IsInducing.codRestrict
| Mathlib/Topology/Constructions.lean | 440 | 444 |
/-
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... | right_cancellation : ∀ {Z : C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h
| Mathlib/CategoryTheory/Category/Basic.lean | 266 | 267 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... |
lemma sqrt_sq : (hA.pow 2 : PosSemidef (A ^ 2)).sqrt = A :=
(hA.pow 2).posSemidef_sqrt.eq_of_sq_eq_sq hA (hA.pow 2).sq_sqrt
include hA in
lemma eq_sqrt_of_sq_eq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B) : A = hB.sqrt := by
subst B
| Mathlib/LinearAlgebra/Matrix/PosDef.lean | 256 | 262 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | [NormedSpace ℝ E'] [CompleteSpace E] [CompleteSpace E'] (hf : Integrable f ν)
(hg : Integrable g μ) : ∫ x, (f ⋆[L, ν] g) x ∂μ = L (∫ x, f x ∂ν) (∫ x, g x ∂μ) := by
refine (integral_integral_swap (by apply hf.convolution_integrand L hg)).trans ?_
simp_rw [integral_comp_comm _ (hg.comp_sub_right _), integral_... | Mathlib/Analysis/Convolution.lean | 859 | 884 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... |
variable {F : Type*} [EquivLike F M N] [StrongHomClass L F M N] (g : F)
@[simp]
theorem realize_boundedFormula (φ : L.BoundedFormula α n) {v : α → M}
{xs : Fin n → M} : φ.Realize (g ∘ v) (g ∘ xs) ↔ φ.Realize v xs := by
| Mathlib/ModelTheory/Semantics.lean | 875 | 880 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Junyan Xu
-/
import Mathlib.Topology.Sheaves.PUnit
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.Functors
/-!
# Skyscraper (pre)sheaves
A skyscra... | SkyscraperPresheafFunctor.map' p₀ f ≫ SkyscraperPresheafFunctor.map' p₀ g := by
ext U
simp only [SkyscraperPresheafFunctor.map'_app, NatTrans.comp_app]
split_ifs with h <;> aesop_cat
/-- Taking skyscraper presheaf at a point is functorial: `c ↦ skyscraper p₀ c` defines a functor by
sending every `f : a ⟶ b... | Mathlib/Topology/Sheaves/Skyscraper.lean | 100 | 107 |
/-
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... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 123 | 127 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | dsimp
simp_rw [negSucc_eq, shiftLeft_neg, Nat.shiftLeft'_false, Nat.shiftRight_add,
← Nat.shiftLeft_sub _ le_rfl, Nat.sub_self, Nat.shiftLeft_zero, ← shiftRight_natCast,
← shiftRight_add', Nat.cast_one]
| -[m+1], n, -[k+1] =>
subNatNat_elim n k.succ
(fun n k i => -[m+1] <... | Mathlib/Data/Int/Bitwise.lean | 390 | 398 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | simp_rw [expand_C] at hf
exact absurd (isUnit_C.2 hr) hf.1
· rw [add_zero, pow_zero, expand_one]
constructor <;> rfl
end CharP
/-- If `n ≠ 0` in `F`, then `X ^ n - a` is separable for any `a ≠ 0`. -/
theorem separable_X_pow_sub_C {n : ℕ} (a : F) (hn : (n : F) ≠ 0) (ha : a ≠ 0) :
Separable (X ^ n - C... | Mathlib/FieldTheory/Separable.lean | 418 | 440 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
/-!
# Quadratic reciprocity.
## Main r... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 173 | 178 | |
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,052 | 1,053 | |
/-
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 | 1,169 | 1,172 | |
/-
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.Order.Monovary
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Product of co... | add_comm _ ((b * b) • f y • g y), add_add_add_comm, add_comm ((a * b) • f y • g x)]
lemma ConcaveOn.smul' [OrderedSMul 𝕜 E] (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
(hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s) :
ConcaveOn 𝕜 s (f • g) := by
refine ⟨hf.1, ... | Mathlib/Analysis/Convex/Mul.lean | 50 | 67 |
/-
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... | Mathlib/Data/Rel.lean | 424 | 425 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Andrew Yang
-/
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Endofunctors as a monoidal category.
We give the monoidal category structure on `C ⥤ C`,
and show that... | (F.map (α_ m₁ m₂ m₃).hom).app X ≫
(δ F m₁ (m₂ ⊗ m₃)).app X ≫ (δ F m₂ m₃).app ((F.obj m₁).obj X) := by
rw [map_associator]
dsimp
simp only [Category.id_comp, Category.assoc, μ_δ_app_assoc, μ_δ_app,
endofunctorMonoidalCategory_tensorObj_obj, Category.comp_id]
| Mathlib/CategoryTheory/Monoidal/End.lean | 253 | 259 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | (bn : b ∣ n) : a = m.gcd (a * b) := by rw [gcd_comm]; apply Coprime.factor_eq_gcd_left cop am bn
theorem Coprime.factor_eq_gcd_right_right {a b m n : ℕ+} (cop : m.Coprime n) (am : a ∣ m)
| Mathlib/Data/PNat/Prime.lean | 245 | 247 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | [NormedSpace 𝕜 E] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G}
{g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} {x : E}
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y : E in 𝓝 x,... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 257 | 295 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
/-!
# Formal multilinear series
In this file we define `FormalMultilinearSeries 𝕜 E F` to be a family o... | theorem norm_apply_eq_norm_coef : ‖p n‖ = ‖coeff p n‖ := by
| Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 311 | 311 |
/-
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... |
@[continuity, fun_prop]
lemma Pi.continuous_restrict (S : Set ι) :
Continuous (S.restrict : (∀ i : ι, π i) → (∀ i : S, π i)) :=
Pi.continuous_precomp' ((↑) : S → ι)
| Mathlib/Topology/Constructions.lean | 674 | 678 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | DecidableRel (SameCycle f⁻¹) := fun x y =>
decidable_of_iff (f.SameCycle x y) (sameCycle_inv).symm
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 207 | 209 |
/-
Copyright (c) 2018 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Sequences in topological spaces
In t... | FrechetUrysohnSpace (Subtype p) :=
IsInducing.subtypeVal.frechetUrysohnSpace
/-- In a sequential space, a set is closed iff it's sequentially closed. -/
theorem isSeqClosed_iff_isClosed [SequentialSpace X] {M : Set X} : IsSeqClosed M ↔ IsClosed M :=
⟨IsSeqClosed.isClosed, IsClosed.isSeqClosed⟩
/-- The preimag... | Mathlib/Topology/Sequences.lean | 168 | 175 |
/-
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
/-!... | induction t with
| zero => simp
| tmul m f => simp [finsuppRight_apply_tmul_apply]
| add x y hx hy => simp [map_add, hx, hy]
@[simp]
| Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 137 | 142 |
/-
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... | def orderIsoUnitIntervalBirational : ℝ≥0∞ ≃o Icc (0 : ℝ) 1 :=
orderIsoIicOneBirational.trans <| (orderIsoIicCoe 1).trans <| (NNReal.orderIsoIccZeroCoe 1).symm
@[simp]
theorem orderIsoUnitIntervalBirational_apply_coe (x : ℝ≥0∞) :
(orderIsoUnitIntervalBirational x : ℝ) = (x⁻¹ + 1)⁻¹.toReal :=
| Mathlib/Data/ENNReal/Inv.lean | 581 | 586 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ t ⊆ s, t.Countable ∧ s ⊆ ⋃ x ∈ t, f x := by
have : ∀ x : s, (↑) ⁻¹' f x ∈ 𝓝 x := fun x => preimage_coe_mem_nhds_subtype.2 (hf x x.2)
rcases countable_cover_nhds this with ⟨t, htc, htU⟩
refine ⟨(↑) '' t, Subtype.coe_image_subset _ _, htc.image _, fun x hx => ?_⟩
simp only [... | Mathlib/Topology/Bases.lean | 843 | 847 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Equiv.Opposite
import Mathlib.Algebra.... | @[simp]
theorem compRight_apply (f : M₂ →ₗ[R] M₃) (g : M →ₗ[R] M₂) : compRight f g = f.comp g :=
rfl
/-- Applying a linear map at `v : M`, seen as a linear map from `M →ₗ[R] M₂` to `M₂`.
See also `LinearMap.applyₗ'` for a version that works with two different semirings.
This is the `LinearMap` version of `toAddMono... | Mathlib/Algebra/Module/LinearMap/End.lean | 374 | 383 |
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.Linea... | intro ⟨σ', hσ'⟩
obtain ⟨n, rfl : σ ^ n = σ'⟩ := mem_powers_iff_mem_zpowers.mpr (hσ σ')
rw [smul_pow', Submonoid.smul_def, AlgEquiv.smul_def, hv', smul_pow, ← pow_mul,
mul_comm, pow_mul, hζ.pow_eq_one, one_pow, one_smul]
· -- Since `σ` does not fix `K⟮α⟯`, `K⟮α⟯` is `L`.
apply IsGalois.intermedia... | Mathlib/FieldTheory/KummerExtension.lean | 483 | 493 |
/-
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.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... | (roots_map F.subtype hf ▸ mem_roots'.mpr ⟨Polynomial.map_ne_zero hnz, hx⟩)
exact x.2
theorem image_rootSet [Algebra R K] [Algebra R L] {p : R[X]} (h : p.Splits (algebraMap R K))
| Mathlib/Algebra/Polynomial/Splits.lean | 357 | 360 |
/-
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` ... | variable {l} {x : α}
theorem mem_of_formPerm_apply_ne (h : l.formPerm x ≠ x) : x ∈ l := by
| Mathlib/GroupTheory/Perm/List.lean | 95 | 97 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Q₁ Q₂`: the quadratic for... |
end Ring
end Pi
| Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 365 | 368 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,546 | 1,548 | |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... | simp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.linearIsometry_apply]
/-- In the canonical isometric isomorphism between a Hilbert sum `E` of `G` and `lp G 2`,
a vector `w : lp G 2` is the image of the infinite sum of the associated elements in `E`, and this
sum indeed converges. -/
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 301 | 305 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Order.ScottTopology
/-!
# Scott Topological Spaces
A type of topological spaces whose notion
of con... |
theorem isωSup_ωSup {α} [OmegaCompletePartialOrder α] (c : Chain α) : IsωSup c (ωSup c) := by
constructor
· apply le_ωSup
| Mathlib/Topology/OmegaCompletePartialOrder.lean | 110 | 113 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... | exact le_iSup₂ (f := fun i (_ : i ∈ S) => m i) i hi'
theorem generateFrom_piiUnionInter_measurableSet (m : ι → MeasurableSpace α) (S : Set ι) :
generateFrom (piiUnionInter (fun n => { s | MeasurableSet[m n] s }) S) = ⨆ i ∈ S, m i := by
refine le_antisymm ?_ ?_
· rw [← @generateFrom_measurableSet α (⨆ i ∈ S, ... | Mathlib/MeasureTheory/PiSystem.lean | 472 | 477 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.NormNum
/-!
# Convex and concave functions
This... | simp_rw [zero_smul, zero_add, one_smul, le_rfl]
obtain rfl | hb' := hb.eq_or_lt
· rw [add_zero] at hab
subst a
simp_rw [zero_smul, add_zero, one_smul, le_rfl]
exact h hx hy ha' hb' hab
theorem concaveOn_iff_forall_pos {s : Set E} {f : E → β} :
ConcaveOn 𝕜 s f ↔
Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s →... | Mathlib/Analysis/Convex/Function.lean | 315 | 328 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... |
theorem preimage_val_eq_preimage_val_iff (s t u : Set α) :
(Subtype.val : s → α) ⁻¹' t = Subtype.val ⁻¹' u ↔ s ∩ t = s ∩ u :=
preimage_coe_eq_preimage_coe_iff
| Mathlib/Data/Set/Image.lean | 1,219 | 1,223 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | refine (Real.iInf_Ioi_eq_iInf_rat_gt _ ?_ f.mono).symm
refine ⟨f x, fun y => ?_⟩
rintro ⟨y, hy_mem, rfl⟩
exact f.mono (le_of_lt hy_mem)
| Mathlib/MeasureTheory/Measure/Stieltjes.lean | 76 | 80 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | @[simp] lemma sum_measure_singleton {s : Finset α} [MeasurableSingletonClass α] :
∑ x ∈ s, μ {x} = μ s := by
trans ∑ x ∈ s, μ (id ⁻¹' {x})
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 214 | 216 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | reprPrec f p :=
| Mathlib/Data/Complex/Basic.lean | 771 | 771 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
/-!
# `min` and `max` in... | section LinearOrderedAddCommGroup
| Mathlib/Algebra/Order/Group/MinMax.lean | 63 | 64 |
/-
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
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... |
lemma Matrix.comp_diagonal (d) :
comp m m n n R (diagonal d) =
(blockDiagonal d).reindex (.prodComm ..) (.prodComm ..) := by
ext
simp [diagonal, blockDiagonal, Matrix.ite_apply]
end Maps
| Mathlib/Data/Matrix/Block.lean | 854 | 861 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
/-!
# Convex hull
This file defines the convex hull of a set `s` in a module. `con... | protected alias ⟨_, Set.Nonempty.convexHull⟩ := convexHull_nonempty_iff
theorem segment_subset_convexHull (hx : x ∈ s) (hy : y ∈ s) : segment 𝕜 x y ⊆ convexHull 𝕜 s :=
(convex_convexHull _ _).segment_subset (subset_convexHull _ _ hx) (subset_convexHull _ _ hy)
@[simp]
theorem convexHull_singleton (x : E) : convex... | Mathlib/Analysis/Convex/Hull.lean | 94 | 100 |
/-
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.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Sub.Basic
impo... |
variable [AddLeftReflectLE R]
| Mathlib/Algebra/Order/Ring/Canonical.lean | 113 | 115 |
/-
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.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... | _ ⟨x₂, hx₂, y₂, hy₂, a₂, b₂, ha₂, hb₂, hab₂, rfl⟩ p q hp hq hpq
rcases hs.exists_mem_add_smul_eq hx₁ hx₂ (mul_nonneg hp ha₁) (mul_nonneg hq ha₂) with ⟨x, hxs, hx⟩
rcases ht.exists_mem_add_smul_eq hy₁ hy₂ (mul_nonneg hp hb₁) (mul_nonneg hq hb₂) with ⟨y, hyt, hy⟩
refine ⟨_, hxs, _, hyt, p * a₁ + q * a₂, p * b₁ ... | Mathlib/Analysis/Convex/Join.lean | 153 | 157 |
/-
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.Order.Preorder.Chain
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-13")
| Mathlib/Order/Chain.lean | 184 | 188 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,150 | 1,153 | |
/-
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 | 1,284 | 1,286 | |
/-
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... | (e₀ : X ≅ X') (e₁ : Y ≅ Y') (e : c.pt ≅ c'.pt)
(comm₁ : e₀.hom ≫ f' = f ≫ e₁.hom := by aesop_cat)
| Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 622 | 623 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | rw [← not_even_iff_odd, even_add, not_iff, ← not_even_iff_odd]
| Mathlib/Algebra/Ring/Parity.lean | 268 | 268 |
/-
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... | section PartialOrder
variable [Preorder α] [PartialOrder β]
| Mathlib/Algebra/Order/Module/Defs.lean | 684 | 685 |
/-
Copyright (c) 2019 Michael Howes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Howes, Newell Jensen
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup.Defs
/-!
# Defining a gr... | mk rels (FreeGroup.of x)
lemma mk_eq_one_iff {rels : Set (FreeGroup α)} {x : FreeGroup α} :
mk rels x = 1 ↔ x ∈ Subgroup.normalClosure rels :=
QuotientGroup.eq_one_iff _
| Mathlib/GroupTheory/PresentedGroup.lean | 53 | 58 |
/-
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.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... |
/-- A code for the identity function. -/
protected def id : Code :=
pair left right
/-- Given a code `c` taking a pair as input, returns a code using `n` as the first argument to `c`.
| Mathlib/Computability/PartrecCode.lean | 103 | 108 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib... | theorem single_zero_smul_eq_smul (Γ) [AddCommMonoid Γ] [PartialOrder Γ] [AddAction Γ Γ']
[IsOrderedCancelVAdd Γ Γ'] [MulZeroClass R] [SMulWithZero R V] {r : R}
{x : HahnModule Γ' R V} :
(HahnSeries.single (0 : Γ) r) • x = r • x := by
ext
exact coeff_single_zero_smul
@[simp]
theorem zero_smul' [Zero R] ... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 311 | 319 |
/-
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... | induction S using Multiset.induction with
| empty => simp
| cons _ _ ih => simp [ih]
| Mathlib/Data/Multiset/Bind.lean | 71 | 73 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 1,534 | 1,537 | |
/-
Copyright (c) 2021 Chris Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Bailey
-/
import Mathlib.Data.Nat.Notation
import Mathlib.Data.String.Defs
import Mathlib.Tactic.Basic
import Batteries.Tactic.Alias
/-!
# Miscellaneous lemmas about strings
-/
| namespace String
| Mathlib/Data/String/Lemmas.lean | 15 | 16 |
/-
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... | exact addRothNumber_union_le _ _
@[simp]
theorem rothNumberNat_zero : rothNumberNat 0 = 0 :=
rfl
theorem addRothNumber_Ico (a b : ℕ) : addRothNumber (Ico a b) = rothNumberNat (b - a) := by
obtain h | h := le_total b a
· rw [Nat.sub_eq_zero_of_le h, Ico_eq_empty_of_le h, rothNumberNat_zero, addRothNumber_empty... | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 429 | 441 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
/-!
# Derivative as the limit of the slope
In this file we relate the ... | ← tendsto_zero_iff_norm_tendsto_zero, slope_def_module, smul_sub]
_ ↔ Tendsto (fun y ↦ slope f x y - (y - x)⁻¹ • (y - x) • f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) :=
.symm <| tendsto_inf_principal_nhds_iff_of_forall_eq <| by simp
_ ↔ Tendsto (fun y ↦ slope f x y - f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) := tendsto_congr' <| by... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 51 | 63 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... |
@[simp]
theorem cycleOf_apply_apply_self (f : Perm α) [DecidableRel f.SameCycle] (x : α) :
cycleOf f x (f x) = f (f x) := by
convert cycleOf_apply_apply_pow_self f x 1 using 1
| Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 101 | 105 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | simpa [H.1] using H.2
lemma coprime_factorizationLCMLeft_factorizationLCMRight :
(factorizationLCMLeft a b).Coprime (factorizationLCMRight a b) := by
rw [factorizationLCMLeft, factorizationLCMRight]
refine coprime_prod_left_iff.mpr fun p hp ↦ coprime_prod_right_iff.mpr fun q hq ↦ ?_
| Mathlib/Data/Nat/Factorization/Basic.lean | 437 | 442 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | sub_self]
have ht (x) (hx₁ : x ∈ t₁) (hx₂ : x ∉ t₂) : w₁ x = 0 := by
simpa [w, hx₁, hx₂] using hs _ (ht₁ hx₁)
refine ⟨w₁, ?_, ?_, ?_⟩
· simp only [and_imp, Finset.mem_inter]
exact fun y hy₁ _ ↦ h₁w₁ y hy₁
all_goals
· rwa [sum_subset inter_subset_left]
rintro x
simp_intro hx₁ hx₂
simp... | Mathlib/Analysis/Convex/Combination.lean | 539 | 557 |
/-
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, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t := by
rcases s with ⟨l⟩
simpa using continuousOn_list_prod l
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_finset_prod {f : ι → X → M} (s : Finset ι) :
| Mathlib/Topology/Algebra/Monoid.lean | 841 | 846 |
/-
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.ObjectProperty.ClosedUnderIsomorphisms
import Mathlib.CategoryTheory.Localization.CalculusOfFractions
import Mathlib.CategoryTheory.Localization.T... | exact ⟨_, _, _, hT, h⟩
lemma isoClosure_W : S.isoClosure.W = S.W := by
ext X Y f
constructor
· rintro ⟨Z, g, h, mem, ⟨Z', hZ', ⟨e⟩⟩⟩
refine ⟨Z', g ≫ e.hom, e.inv ≫ h, isomorphic_distinguished _ mem _ ?_, hZ'⟩
exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) e.symm
| Mathlib/CategoryTheory/Triangulated/Subcategory.lean | 145 | 152 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberThe... | /-- The Dirichlet character mod `1` corresponds to the constant function `1`. -/
lemma modOne_eq_one {R : Type*} [CommMonoidWithZero R] {χ : DirichletCharacter R 1} :
((χ ·) : ℕ → R) = 1 := by
ext
rw [χ.level_one, MulChar.one_apply (isUnit_of_subsingleton _), Pi.one_apply]
lemma LSeries_modOne_eq : L ↗χ₁ = L 1... | Mathlib/NumberTheory/LSeries/Dirichlet.lean | 155 | 165 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... |
theorem sublistsLen_of_length_lt {n} {l : List α} (h : l.length < n) : sublistsLen n l = [] :=
eq_nil_iff_forall_not_mem.mpr fun _ =>
mem_sublistsLen.not.mpr fun ⟨hs, hl⟩ => (h.trans_eq hl.symm).not_le (Sublist.length_le hs)
@[simp]
theorem sublistsLen_length : ∀ l : List α, sublistsLen l.length l = [l]
| Mathlib/Data/List/Sublists.lean | 286 | 292 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... |
lemma HomologyData.exact_iff' (h : S.HomologyData) :
S.Exact ↔ IsZero h.right.H := by
haveI := HasHomology.mk' h
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 93 | 96 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
/-!
# The rational numbers form a linear ordered field
This f... | Mathlib/Algebra/Order/Ring/Rat.lean | 186 | 189 | |
/-
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.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Mathlib/Data/Finsupp/Basic.lean | 1,584 | 1,591 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Pi.Basic
/-!
# Dependent-typed matrices
-/
universe u u' v w z
/-- `DMatrix m n` is the type of dep... | map_add' := DMatrix.map_add f
@[simp]
theorem AddMonoidHom.mapDMatrix_apply [∀ i j, AddMonoid (α i j)] {β : m → n → Type w}
| Mathlib/Data/Matrix/DMatrix.lean | 154 | 157 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | μ[∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun ... | Mathlib/Probability/Martingale/Upcrossing.lean | 397 | 406 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... |
variable [NormedDivisionRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
| Mathlib/Analysis/LocallyConvex/Basic.lean | 150 | 152 |
/-
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.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | HomologicalComplex.homologyMap φ 0 ≫ L.isoHomologyπ₀.inv =
K.isoHomologyπ₀.inv ≫ HomologicalComplex.cyclesMap φ 0 := by
simp only [← cancel_epi (K.homologyπ 0), HomologicalComplex.homologyπ_naturality_assoc,
HomologicalComplex.isoHomologyπ_hom_inv_id, comp_id,
HomologicalComplex.isoHomologyπ_hom_inv... | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 683 | 696 |
/-
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.Data.Set.Lattice
/-! # Semiquotients
A data type for semiquotients, which are classically equivalent to
nonempty sets, but are useful for programming... | Mathlib/Data/Semiquot.lean | 234 | 238 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
/-! # Adjoi... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 132 | 136 | |
/-
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.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.RingTheory.AlgebraTower
import Mathli... | exact .inr h
theorem finite_of_finrank_pos (h : 0 < finrank R M) : Module.Finite R M := by
contrapose h
simp [finrank_of_not_finite h]
| Mathlib/LinearAlgebra/Dimension/Free.lean | 180 | 185 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | _ = x ⊓ y \ x ⊓ x \ y := by rw [inf_inf_sdiff, bot_sup_eq]
_ = x ⊓ x \ y ⊓ y \ x := by ac_rfl
_ = x \ y ⊓ y \ x := by rw [inf_of_le_right sdiff_le']
theorem disjoint_sdiff_sdiff : Disjoint (x \ y) (y \ x) :=
disjoint_iff_inf_le.mpr sdiff_inf_sdiff.le
@[simp]
theorem inf_sdiff_self_right : x ⊓ y \ ... | Mathlib/Order/BooleanAlgebra.lean | 149 | 160 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Semiconj.Basic
/-!
# Additional lemmas about commuting pairs of elements in... |
@[to_additive]
protected theorem div_mul_div_comm (hbd : Commute b d) (hbc : Commute b⁻¹ c) :
a / b * (c / d) = a * c / (b * d) := by
| Mathlib/Algebra/Group/Commute/Basic.lean | 54 | 57 |
/-
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.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... | refine ContinuousOn.aestronglyMeasurable (fun t ht ↦ ?_) measurableSet_Ioi
exact (Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr (hc.trans ht).ne')).continuousWithinAt
| Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 182 | 184 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
/-!
# Basis on a quaternion-like algebra
## Main definitions
* `QuaternionAlgebra.Basis A c₁ c₂ c₃`: a basis... | smul_mul_assoc, mul_assoc, j_mul_j]
linear_combination (norm := module)
| Mathlib/Algebra/QuaternionBasis.lean | 104 | 106 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | exact (-o).norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two h
/-- An angle in a right-angled triangle expressed using `arccos`, version subtracting vectors. -/
theorem oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle y (y - x) = Real.arccos (‖y‖ / ‖y - x‖... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 238 | 242 |
/-
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.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `𝒜 : Finset (Fins... | simp only [mem_memberSubfamily, mem_image]
refine ⟨?_, fun hs ↦ ⟨⟨s, hs, rfl⟩, h𝒜 _ hs⟩⟩
rintro ⟨⟨t, ht, hts⟩, hs⟩
rwa [← insert_erase_invOn.2.injOn (h𝒜 _ ht) hs hts]
| Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 126 | 129 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | Mathlib/Algebra/GroupWithZero/Basic.lean | 500 | 501 | |
/-
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.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 501 | 508 |
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