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) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | [TopologicalSpace H₃] [ChartedSpace H₁ H₂] [ChartedSpace H₂ H₃] {G₁ : StructureGroupoid H₁}
[HasGroupoid H₂ G₁] [ClosedUnderRestriction G₁] (G₂ : StructureGroupoid H₂) [HasGroupoid H₃ G₂]
theorem HasGroupoid.comp
(H : ∀ e ∈ G₂, LiftPropOn (IsLocalStructomorphWithinAt G₁) (e : H₂ → H₂) e.source) :
@HasGroup... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 648 | 666 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 763 | 766 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Probability measures
Th... |
theorem normalize_testAgainstNN (nonzero : μ ≠ 0) (f : Ω →ᵇ ℝ≥0) :
μ.normalize.toFiniteMeasure.testAgainstNN f = μ.mass⁻¹ * μ.testAgainstNN f := by
| Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 443 | 445 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | LeftInvOn (fun x ↦ (g₁ x.1, g₂ x.2)) (fun x ↦ (f₁ x.1, f₂ x.2)) (s₁ ×ˢ s₂) :=
fun _x hx ↦ Prod.ext (h₁ hx.1) (h₂ hx.2)
lemma RightInvOn.prodMap (h₁ : RightInvOn g₁ f₁ t₁) (h₂ : RightInvOn g₂ f₂ t₂) :
RightInvOn (fun x ↦ (g₁ x.1, g₂ x.2)) (fun x ↦ (f₁ x.1, f₂ x.2)) (t₁ ×ˢ t₂) :=
| Mathlib/Data/Set/Function.lean | 1,220 | 1,224 |
/-
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... | subst h; exact mono_of_mono f g
theorem epi_of_epi {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [Epi (f ≫ g)] : Epi g :=
⟨fun _ _ w => (cancel_epi (f ≫ g)).1 <| by simp only [Category.assoc, w]⟩
theorem epi_of_epi_fac {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [Epi h]
| Mathlib/CategoryTheory/Category/Basic.lean | 325 | 330 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | theorem IsUniformInducing.of_comp_iff {g : β → γ} (hg : IsUniformInducing g) {f : α → β} :
IsUniformInducing (g ∘ f) ↔ IsUniformInducing f := by
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 67 | 68 |
/-
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.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
/-!
# Complex roots of unity
In this file we show that th... | · rintro ⟨i, -, hi, rfl⟩; exact isPrimitiveRoot_exp_of_coprime i n hn hi
intro h
have : NeZero n := ⟨hn⟩
obtain ⟨i, hi, rfl⟩ :=
(isPrimitiveRoot_exp n hn).eq_pow_of_pow_eq_one h.pow_eq_one
refine ⟨i, hi, ((isPrimitiveRoot_exp n hn).pow_iff_coprime (Nat.pos_of_ne_zero hn) i).mp h, ?_⟩
rw [← exp_nat_mul]
... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 58 | 69 |
/-
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... | (shiftIso F m a' a'' ha'').hom.app X := by
simp [F.shiftIso_add n m a a' a'' ha' ha'']
lemma shiftIso_add_inv_app (n m a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
(F.shiftIso (m + n) a a'' (by rw [add_assoc, ha', ha''])).inv.app X =
(shiftIso F m a' a'' ha'').inv.app X ≫
| Mathlib/CategoryTheory/Shift/ShiftSequence.lean | 148 | 153 |
/-
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... | top_le_sup_compl a := a.or_not_self.ge
| Mathlib/Order/BooleanAlgebra.lean | 741 | 742 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... |
lemma j_eq_zero (h : W.c₄ = 0) : W.j = 0 := by
rw [j_eq_zero_iff', h, zero_pow three_ne_zero]
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 384 | 387 |
/-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Coset.Basic
/-!
# Double cosets
This file defines double cosets for two subgroups `H K` o... | rw [rel_iff, QuotientGroup.rightRel_apply]
constructor
· rintro ⟨b, hb, a, rfl : a = 1, rfl⟩
rwa [mul_one, mul_inv_cancel_right]
· rintro (h : b * a⁻¹ ∈ H)
exact ⟨b * a⁻¹, h, 1, rfl, by rw [mul_one, inv_mul_cancel_right]⟩
/-- Create a doset out of an element of `H \ G / K` -/
def quotToDoset (H K : Sub... | Mathlib/GroupTheory/DoubleCoset.lean | 93 | 102 |
/-
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... | simp only [negSucc_eq, ← Int.natCast_succ, bodd_neg, bodd_coe, Nat.bodd_mul]
theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
| (n : ℕ) => by
rw [show (cond (bodd n) 1 0 : ℤ) = (cond (bodd n) 1 0 : ℕ) by cases bodd n <;> rfl]
| Mathlib/Data/Int/Bitwise.lean | 145 | 149 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,164 | 1,166 | |
/-
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.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | rw [Icc_eq_cons_Ico h, prod_cons]
@[to_additive]
| Mathlib/Algebra/BigOperators/Intervals.lean | 30 | 32 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | apply Set.mem_range_self
apply h (Set.range a)
· use a 37
apply Set.mem_range_self
· rintro x ⟨m, hm⟩ y ⟨n, hn⟩
use m ⊔ n
rw [← hm, ← hn]
apply RelHomClass.map_sup a
theorem isSupFiniteCompact_iff_all_elements_compact :
IsSupFiniteCompact α ↔ ∀ k : α, IsCompactElement k :=... | Mathlib/Order/CompactlyGenerated/Basic.lean | 227 | 245 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
/-!
# Extending an alternating map to the exterior algebra
## Main de... | invFun F i := F.compAlternatingMap (ιMulti R i)
left_inv _ := funext fun _ => liftAlternating_comp_ιMulti _
right_inv F :=
(liftAlternating_comp _ _).trans <| by rw [liftAlternating_ιMulti, LinearMap.comp_id]
/-- To show that two linear maps from the exterior algebra agree, it suffices to show they agree on
... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 138 | 147 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | theorem to_real_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]
| Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 81 | 81 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | ⟨range_nonempty f, isPreconnected_range h⟩
theorem Function.Surjective.connectedSpace [ConnectedSpace α] [TopologicalSpace β]
{f : α → β} (hf : Surjective f) (hf' : Continuous f) : ConnectedSpace β := by
rw [connectedSpace_iff_univ, ← hf.range_eq]
exact isConnected_range hf'
instance Quotient.instConnectedS... | Mathlib/Topology/Connected/Basic.lean | 641 | 652 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | have hx := congr_arg (Polynomial.aeval x) h.symm
rw [minpoly.aeval, map_sub, map_pow, aeval_X, aeval_C, sub_eq_zero, eq_comm] at hx
use n
rw [h, Polynomial.map_sub, Polynomial.map_pow, map_X, map_C, hx, map_pow,
| Mathlib/FieldTheory/SeparableDegree.lean | 668 | 671 |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
/-!
# Diagonal matrices
This file co... | intro i j h
simp [ha h]
@[simp]
| Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 92 | 95 |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | · interval_cases n <;> simp only [Nat.succ_eq_add_one] <;> ring_nf <;> try simp only [coeffs]
· rw [coeff_eq_zero hn, (degree_le_iff_coeff_zero (f : R[X]) 3).mp f.2]
simpa using hn
| Mathlib/Algebra/CubicDiscriminant.lean | 250 | 252 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Relations holding pairwise
This file develops pairwise relations ... | forall_mem_image.2 fun _x hx ↦ forall_mem_image.2 fun _y hy hne ↦ h hx hy <| ne_of_apply_ne _ hne
| Mathlib/Data/Set/Pairwise/Basic.lean | 193 | 193 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | @[simp] theorem replicateRow_eq_zero [Zero α] [Nonempty ι] (v : n → α) :
replicateRow ι v = 0 ↔ v = 0 :=
replicateRow_inj
| Mathlib/Data/Matrix/RowCol.lean | 112 | 114 |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | Mathlib/NumberTheory/FermatPsp.lean | 371 | 375 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.UrysohnsBounded
/-!
# Tietze extension theore... | such that the distance between `g ∘ e` and `f` is at most `(2 / 3) * ‖f‖`. -/
theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : IsClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖ := by
have h3 : (0 : ℝ) < 3 := by norm_num1
have h23 : 0 < (2 / 3 : ℝ) := by... | Mathlib/Topology/TietzeExtension.lean | 169 | 213 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 858 | 859 | |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
/-!
#... | simpa only [Ne, map_zero] using
(IsFractionRing.injective (𝕎 k) (FractionRing (𝕎 k))).ne hq'''
rw [zpow_sub₀ (FractionRing.p_nonzero p k)]
field_simp [FractionRing.p_nonzero p k]
convert congr_arg (fun x => algebraMap (𝕎 k) (FractionRing (𝕎 k)) x) key using 1
· simp only [RingHom.map_mul, RingHom.... | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 241 | 267 |
/-
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.Group.Int.Defs
import Mathlib.Algebra.Order.Monoid.Defs
/-!
# The integers form a linear ordered group
This file contains the instance necessar... | Mathlib/Algebra/Order/Group/Int.lean | 88 | 89 | |
/-
Copyright (c) 2023 Shogo Saito. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shogo Saito. Adapted for mathlib by Hunter Monroe
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.ChineseRemainder
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Nat.Pai... | -/
namespace Nat
lemma coprime_mul_succ {n m a} (ha : m - n ∣ a) : Coprime (n * a + 1) (m * a + 1) :=
Nat.coprime_of_dvd fun p pp hn hm => by
have : p ∣ (m - n) * a := by
simpa [Nat.succ_sub_succ, ← Nat.mul_sub_right_distrib] using
Nat.dvd_sub hm hn
have : p ∣ a := by
rcases (Nat.Prime.d... | Mathlib/Logic/Godel/GodelBetaFunction.lean | 45 | 55 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
/-!
# Extending an alternating map to the exterior algebra
## Main de... | ext v
dsimp
induction v using CliffordAlgebra.left_induction with
| algebraMap => rw [liftAlternating_algebraMap, ιMulti_zero_apply, Algebra.algebraMap_eq_smul_one]
| add _ _ hx hy => rw [map_add, hx, hy]
| ι_mul _ _ hx => simp_rw [liftAlternating_ι_mul, ιMulti_succ_curryLeft, liftAlternating_comp,
Li... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 124 | 134 |
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Topolog... | (hB : ∀ᶠ x in l, f x ≤ B) (hg : Tendsto g l (𝓝 0)) :
Tendsto (fun x ↦ f x * g x) l (𝓝 0) := by
set C := max |b| |B|
have hbC : -C ≤ b := neg_le.mpr (le_max_of_le_left (neg_le_abs b))
have hBC : B ≤ C := le_max_of_le_right (le_abs_self B)
apply bdd_le_mul_tendsto_zero' C _ hg
| Mathlib/Topology/Algebra/Order/Field.lean | 228 | 233 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | sup_inf_sdiff s t
| Mathlib/Data/Set/Basic.lean | 1,124 | 1,125 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Big operators for `NatAntidiagonal`
This file contains theor... | (∑ p ∈ antidiagonal (n + 1), f p) = f (n + 1, 0) + ∑ p ∈ antidiagonal n, f (p.1, p.2 + 1) :=
@prod_antidiagonal_succ' (Multiplicative N) _ _ _
@[to_additive]
| Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 42 | 45 |
/-
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... | toFun := Associates.out
invFun := Associates.mk
left_inv := Associates.mk_out
right_inv _ := (Associates.out_mk _).trans <| normalize_eq _
map_mul' := Associates.out_mul
end UniqueUnit
section IsDomain
variable [CommRing α] [IsDomain α] [NormalizedGCDMonoid α]
theorem gcd_eq_of_dvd_sub_right {a b c : α} (... | Mathlib/Algebra/GCDMonoid/Basic.lean | 863 | 875 |
/-
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... | · rw [splitMin_eq]
· obtain ⟨v, e⟩ := Valid'.eraseMin_aux hr
| Mathlib/Data/Ordmap/Ordset.lean | 419 | 420 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | exact h ⟨j, hj⟩ hpj
· exact Option.not_mem_none _ hi
· rw [h] at hi
dsimp at hi
obtain rfl := Option.some_inj.1 hi
exact find_min h (show (⟨j, lt_trans hj k.2⟩ : Fin n) < k from hj) hpj
theorem find_min' {p : Fin n → Prop} [DecidablePred p] {i : Fin n} (h : i ∈ Fin.find p) {j : Fin ... | Mathlib/Data/Fin/Tuple/Basic.lean | 1,049 | 1,064 |
/-
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... | Irreducible (X ^ n - C a) ↔ (∀ p : ℕ, p.Prime → p ∣ n → ∀ b : K, b ^ p ≠ a) :=
⟨fun e _ hp hpn ↦ pow_ne_of_irreducible_X_pow_sub_C e hpn hp.ne_one,
X_pow_sub_C_irreducible_of_odd hn⟩
theorem X_pow_sub_C_irreducible_iff_of_odd {n : ℕ} (hn : Odd n) {a : K} :
Irreducible (X ^ n - C a) ↔ (∀ d, d ∣ n → d ≠ 1 ... | Mathlib/FieldTheory/KummerExtension.lean | 122 | 128 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.Algebra.Category.ModuleCat.EpiMono
/-!
# Pseudoelements in ab... | -/
section
| Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | 193 | 197 |
/-
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.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | exact fun ⟨_, _, _, b, hb, _⟩ => ⟨b, hb⟩
@[simp]
theorem disjSups_empty_left : ∅ ○ t = ∅ := by simp [disjSups]
@[simp]
theorem disjSups_empty_right : s ○ ∅ = ∅ := by simp [disjSups]
| Mathlib/Data/Finset/Sups.lean | 444 | 450 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Antidiag.Prod
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.NatAntidiagonal
/-!
# Antidiagonals in ℕ × ℕ as finset... | (Embedding.prodMap ⟨Nat.succ, Nat.succ_injective⟩ (Embedding.refl _)))
(by simp) := by
apply eq_of_veq
rw [cons_val, map_val]
apply Multiset.Nat.antidiagonal_succ
theorem antidiagonal_succ' (n : ℕ) :
antidiagonal (n + 1) =
cons (n + 1, 0)
| Mathlib/Data/Finset/NatAntidiagonal.lean | 67 | 75 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
/-!
# Volume forms and measures on inner product spa... | theorem PiLp.volume_preserving_equiv_symm : MeasurePreserving (WithLp.equiv 2 (ι → ℝ)).symm :=
(EuclideanSpace.volume_preserving_measurableEquiv ι).symm
lemma volume_euclideanSpace_eq_dirac [IsEmpty ι] :
(volume : Measure (EuclideanSpace ℝ ι)) = Measure.dirac 0 := by
rw [← ((EuclideanSpace.volume_preserving_me... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 138 | 143 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Graded tensor products over graded algebras
The graded tens... | ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) := by
rw [tmul_coe_mul_coe_tmul, mul_zero, uzpow_zero, one_smul]
theorem tmul_one_mul_coe_tmul {i₂ : ι} (a₁ : A) (a₂ : 𝒜 i₂) (b₂ : B) :
| Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 207 | 210 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | KaehlerDifferential.derivationQuotKerTotal R S x = 1𝖣x :=
rfl
theorem KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination :
(KaehlerDifferential.derivationQuotKerTotal R S).liftKaehlerDifferential.comp
(Finsupp.linearCombination S (KaehlerDifferential.D R S)) =
Submodule.mkQ... | Mathlib/RingTheory/Kaehler/Basic.lean | 556 | 563 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | rw [← encard_union_eq disjoint_compl_right, union_compl_self]
end Lattice
| Mathlib/Data/Set/Card.lean | 243 | 245 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | theorem _root_.Sbtw.oangle_sign_eq_left {p₁ p₂ p₃ : P} (p₄ : P) (h : Sbtw ℝ p₁ p₂ p₃) :
(∡ p₁ p₄ p₂).sign = (∡ p₁ p₄ p₃).sign :=
h.wbtw.oangle_sign_eq_of_ne_left _ h.left_ne
/-- Given three points in weak order on the same line, with the second not equal to the third,
and a fourth point, the angles at the fourth... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 687 | 763 |
/-
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... | right_inv f := AlternatingMap.ext fun _ => f.congr_arg <| funext fun _ => e.apply_symm_apply _
@[simp]
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 558 | 560 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | theorem LinearMap.toMatrix_basis_equiv [Fintype l] [DecidableEq l] (b : Basis l R M₁)
(b' : Basis l R M₂) :
LinearMap.toMatrix b' b (b'.equiv b (Equiv.refl l) : M₂ →ₗ[R] M₁) = 1 := by
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 582 | 584 |
/-
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... | by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (hl.hasFDerivWithinAt.comp_hasDerivWithinAt x hf.hasDerivWithinAt hs).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx]
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 382 | 385 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,906 | 1,915 | |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | theorem exists_nonneg_left (h : SameRay R x y) (hx : x ≠ 0) : ∃ r : R, 0 ≤ r ∧ r • x = y := by
obtain rfl | hy := eq_or_ne y 0
· exact ⟨0, le_rfl, zero_smul _ _⟩
| Mathlib/LinearAlgebra/Ray.lean | 577 | 579 |
/-
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... | rintro p ⟨n, a, rfl⟩
| Mathlib/Algebra/Polynomial/Basic.lean | 731 | 731 |
/-
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.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | two consecutive arrows are zero. -/
structure IsComplex : Prop where
/-- the composition of two consecutive arrows is zero -/
zero (i : ℕ) (hi : i + 2 ≤ n := by omega) :
| Mathlib/Algebra/Homology/ExactSequence.lean | 72 | 75 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
/-!
# Operator norm: Cartesian products
Interaction of operator norm wit... | /-- The operator norm of the second projection `E × F → F` is at most 1. (It is 0 if `F` is zero, so
the inequality cannot be improved without further assumptions.) -/
lemma norm_snd_le : ‖snd 𝕜 E F‖ ≤ 1 :=
opNorm_le_bound _ zero_le_one (fun ⟨e, f⟩ ↦ by simpa only [one_mul] using le_max_right ‖e‖ ‖f‖)
| Mathlib/Analysis/NormedSpace/OperatorNorm/Prod.lean | 34 | 37 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Rodriguez
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Data.Nat.Choose.Basic
/-... | /-- This lemma was changed on 2024/08/29, the old statement is available
in `Nat.choose_le_pow_div`. -/
lemma choose_le_pow (n k : ℕ) : n.choose k ≤ n ^ k :=
(choose_le_descFactorial n k).trans (descFactorial_le_pow n k)
-- horrific casting is due to ℕ-subtraction
| Mathlib/Data/Nat/Choose/Bounds.lean | 41 | 46 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | map f xs = (mapAccumr (fun x (_ : Unit) ↦ ((), f x)) xs ()).snd := by
induction xs using Vector.revInductionOn <;> simp_all
/--
If there is a set of states that is closed under `f`, and such that `f` produces that same output
for all states in this set, then the state is not actually needed.
Hence, then we can r... | Mathlib/Data/Vector/MapLemmas.lean | 234 | 241 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Tactic.Lift... | | 0, hn => (hn rfl).elim
| 1, _ => by simpa
| Nat.succ <| Nat.succ n, _ => by
simpa only [pow_succ] using (hf.pow_const n.succ_ne_zero).mul' hf
| Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 136 | 140 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite.Range
import Mathlib.Order.Partition.Finpartition
/-!
# Equivalenc... | instance Partition.le : LE (Subtype (@IsPartition α)) :=
⟨fun x y => mkClasses x.1 x.2.2 ≤ mkClasses y.1 y.2.2⟩
/-- Defining a partial order on partitions as the partial order on their induced
equivalence relations. -/
instance Partition.partialOrder : PartialOrder (Subtype (@IsPartition α)) where
le := (· ≤ ·... | Mathlib/Data/Setoid/Partition.lean | 237 | 243 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | |exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) := by
simp only [expNear, mul_zero, add_zero]
| Mathlib/Data/Complex/Exponential.lean | 575 | 576 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
/-!
# The symmetric monoid... | @[simp]
theorem hexagon_forward (X Y Z : ModuleCat.{u} R) :
(α_ X Y Z).hom ≫ (braiding X _).hom ≫ (α_ Y Z X).hom =
(braiding X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (braiding X Z).hom := by
| Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 49 | 52 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,741 | 1,747 | |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Nat
import Mathli... | (castOrderEmbedding (α := ℤ)).image_Icc (by simp [ordConnected_Ici]) a b
| Mathlib/Data/Nat/Cast/SetInterval.lean | 28 | 29 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | toFun := ghostFun
invFun x := mk p fun n => aeval x (xInTermsOfW p R n)
| Mathlib/RingTheory/WittVector/Basic.lean | 207 | 208 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | p.closedBall (x₁ : E) r₁ + p.closedBall (x₂ : E) r₂ ⊆ p.closedBall (x₁ + x₂) (r₁ + r₂) := by
rintro x ⟨y₁, hy₁, y₂, hy₂, rfl⟩
| Mathlib/Analysis/Seminorm.lean | 707 | 708 |
/-
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... | rcases hf η ηpos.ne' int_ff'.1 with ⟨g, hg, Pg⟩
rcases hf' η ηpos.ne' int_ff'.2 with ⟨g', hg', Pg'⟩
refine ⟨g + g', ?_, h1P g g' Pg Pg'⟩
convert (hη _ _ (f.aestronglyMeasurable.sub (h2P g Pg))
(f'.aestronglyMeasurable.sub (h2P g' Pg')) hg hg').le using 2
rw [SimpleFunc.coe_add]
abel
s... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 911 | 921 |
/-
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... | In particular, this version only requires us to show
that `motive` is closed under addition of nontrivial monomials not present in the support.
-/
| Mathlib/Algebra/MvPolynomial/Basic.lean | 379 | 381 |
/-
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... | theorem lt_iff : ∀ {x y : Bool}, x < y ↔ x = false ∧ y = true := by decide
| Mathlib/Data/Bool/Basic.lean | 158 | 159 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.ULift
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Tactic.Ring
/-!
# The characteristic predicate of tensor product
## Main ... | LinearMap.ext (h.lift_eq g)
end
| Mathlib/RingTheory/IsTensorProduct.lean | 175 | 177 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes, Antoine Chambert-Loir
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator.Basic
import Mathlib.GroupTheory.Coset.Basic
import Mathlib.GroupThe... |
variable {G} in
| Mathlib/GroupTheory/Abelianization.lean | 53 | 54 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Equiv
import Mathlib.... | ⟨ofInj f hf⟩
@[simp]
theorem nonempty_encodable : Nonempty (Encodable α) ↔ Countable α :=
⟨fun ⟨h⟩ => @Encodable.countable α h, fun h => ⟨@ofCountable _ h⟩⟩
| Mathlib/Logic/Encodable/Basic.lean | 384 | 389 |
/-
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 | 1,434 | 1,436 | |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Floris van Doorn
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Basic
/-!
# Explicit least witnesses to existentials on positive natural numbers
Implement... | PNat.find h = PNat.find h₂ + 1 := by
| Mathlib/Data/PNat/Find.lean | 101 | 101 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,931 | 1,937 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Probability measures
Th... | exact h s s_mble
theorem eq_of_forall_apply_eq (μ ν : ProbabilityMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → μ s = ν s) : μ = ν := by
ext1 s s_mble
simpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((↑) : ℝ≥0 → ℝ≥0∞) (h s s_mble)
| Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 207 | 212 |
/-
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... | simp_rw [← star_mulVec, hv', mul_add, ← RCLike.real_smul_eq_coe_mul, ← smul_dotProduct]
congr 2 with i
simp only [Pi.star_apply, Pi.smul_apply, RCLike.real_smul_eq_coe_mul, star_mul',
RCLike.star_def, RCLike.conj_ofReal]
| Mathlib/LinearAlgebra/Matrix/PosDef.lean | 239 | 242 |
/-
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... |
/-- Lift an algebra homomorphism that maps polynomials `φ : K[X] →ₐ[S] R[X]`
| Mathlib/FieldTheory/RatFunc/Basic.lean | 607 | 608 |
/-
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... | Int.negOnePow_add, Int.negOnePow_one, Units.val_neg, Int.cast_neg, neg_mul, neg_neg, ih2,
Int.negOnePow_def 2]
norm_cast
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 149 | 151 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ℕ∞`
In this ... | rw [← lift_ofENat.{u, v}, lift_inj]
@[simp]
lemma range_ofENat : range ofENat = Iic ℵ₀ := by
refine (range_subset_iff.2 ofENat_le_aleph0).antisymm fun x (hx : x ≤ ℵ₀) ↦ ?_
rcases hx.lt_or_eq with hlt | rfl
· lift x to ℕ using hlt
| Mathlib/SetTheory/Cardinal/ENat.lean | 142 | 148 |
/-
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.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... | iIndep m μ ↔ iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) μ := by
simp only [iIndep, iIndepSets, Kernel.iIndep]
| Mathlib/Probability/Independence/Basic.lean | 161 | 162 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
import Mathlib.CategoryTheory.Sites.Lo... | extensiveTopology.presheafIsLocallySurjective_iff _ f.val
attribute [local instance] Types.instFunLike Types.instConcreteCategory in
lemma regularTopology.isLocallySurjective_sheaf_of_types [Preregular C] [FinitaryPreExtensive C]
{F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProduc... | Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean | 108 | 138 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-! # Power funct... | -- let icα ← mk_instance_cache α
-- let icβ ← mk_instance_cache β
| Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 263 | 264 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
/-!
# First-Orde... | Mathlib/ModelTheory/Substructures.lean | 987 | 989 | |
/-
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 ... | apply Coprime.gcd_mul_right_cancel a
apply Coprime.coprime_dvd_left bn cop.symm
theorem Coprime.factor_eq_gcd_right {a b m n : ℕ+} (cop : m.Coprime n) (am : a ∣ m) (bn : b ∣ n) :
| Mathlib/Data/PNat/Prime.lean | 238 | 241 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpac... | theorem support_subset_source : support f ⊆ (chartAt H c).source := by
rw [f.support_eq_inter_preimage, ← extChartAt_source I]; exact inter_subset_left
| Mathlib/Geometry/Manifold/BumpFunction.lean | 131 | 132 |
/-
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.Functor.KanExtension.Basic
import Mathlib.CategoryTheory.Localization.Predicate
/-!
# Right derived functors
In this file, given a functor `F : ... | lemma isRightDerivedFunctor_iff_of_iso (α' : F ⟶ L ⋙ RF') (W : MorphismProperty C)
[L.IsLocalization W] (e : RF ≅ RF') (comm : α ≫ whiskerLeft L e.hom = α') :
RF.IsRightDerivedFunctor α W ↔ RF'.IsRightDerivedFunctor α' W := by
simp only [isRightDerivedFunctor_iff_isLeftKanExtension]
exact isLeftKanExtension... | Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean | 66 | 70 |
/-
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.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | a ∈ rfindOpt f ↔ ∃ n, a ∈ f n :=
⟨rfindOpt_spec, fun ⟨n, h⟩ => by
have h' := rfindOpt_dom.2 ⟨_, _, h⟩
obtain ⟨k, hk⟩ := rfindOpt_spec ⟨h', rfl⟩
have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk)
simp at this; simp [this, get_mem]⟩
| Mathlib/Computability/Partrec.lean | 144 | 150 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... | theorem rootsExpandToRoots_apply (x) : (rootsExpandToRoots p f x : R) = x ^ p := rfl
open scoped Classical in
/-- If `f` is a polynomial over an integral domain `R` of characteristic `p`, then there is
| Mathlib/FieldTheory/Perfect.lean | 339 | 342 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | Mathlib/Data/Real/GoldenRatio.lean | 238 | 243 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | contrapose! hm₂
exact mem_erase_of_ne_of_mem hm₂ hm₁
simp [this, h0]
_ ≤ ∑ m ∈ c.divisors.erase d, φ m := by
refine sum_le_sum fun m hm => ?_
have hmc : m ∣ c := by
simp only [mem_erase, mem_divisors] at hm
| Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 501 | 507 |
/-
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... | HasSum (fun n : ℤ ↦ Gammaℝ (s + 1) * (-I) * Int.sign n *
cexp (2 * π * I * a * n) / (↑|n| : ℂ) ^ s / 2) (completedSinZeta a s) := by
let c (n : ℤ) : ℂ := -I * cexp (2 * π * I * a * n) / 2
have hc (n : ℤ) : ‖c n‖ = 1 / 2 := by
| Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 387 | 390 |
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Cofinal
import Mathlib.Order.UpperLower.Principal... | theorem coe_top : ((⊤ : Ideal P) : Set P) = univ :=
rfl
theorem isProper_of_ne_top (ne_top : I ≠ ⊤) : IsProper I :=
⟨fun h ↦ ne_top <| ext h⟩
| Mathlib/Order/Ideal.lean | 191 | 195 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
theorem ncard_le_ncard_insert (a : α) (s : Set α) : s.ncard ≤ (insert a s).ncard := by
classical
| Mathlib/Data/Set/Card.lean | 616 | 618 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... |
theorem mk_mem_col_iff {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ.col j ↔ (i, j) ∈ μ := by simp [col]
protected theorem exists_not_mem_col (μ : YoungDiagram) (j : ℕ) : ∃ i, (i, j) ∉ μ.cells := by
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 307 | 310 |
/-
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... | /-- A rank `n` free module has a basis indexed by `Fin n`. -/
noncomputable def finBasisOfFinrankEq [Module.Finite R M] {n : ℕ} (hn : finrank R M = n) :
Basis (Fin n) R M := (finBasis R M).reindex (finCongr hn)
variable {R M}
| Mathlib/LinearAlgebra/Dimension/Free.lean | 202 | 206 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex
import Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex
import Mathlib.AlgebraicTopology.CechNerve
import Mathlib.A... | @[simp]
theorem shiftFun_zero {n : ℕ} {X : Type*} [Zero X] (f : Fin n → X) : shiftFun f 0 = 0 :=
rfl
@[deprecated (since := "2025-04-19")]
alias shiftFun_0 := shiftFun_zero
| Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | 160 | 166 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | ⟨(extChartAt I x).map_source (mem_extChartAt_source _), by rwa [extChartAt_to_inv], by
rw [extChartAt_to_inv]; apply mem_extChartAt_source⟩
rw [insert_eq_of_mem hxs'] at hmem hsub
-- Then `(extChartAt I x).symm '' v` is the neighborhood we are looking for.
refine ⟨(extChartAt I x).symm '' v, ?_, ?_⟩
·... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 796 | 804 |
/-
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
... | · simp [hu]
rw [derivWithin_zero_of_not_differentiableWithinAt hv, zero_mul,
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 254 | 255 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | -- Note the `'` swapped around due to a move to std4.
| Mathlib/Data/Fin/Basic.lean | 458 | 459 |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... |
open Function in
@[simp]
| Mathlib/Data/Option/Basic.lean | 330 | 332 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Data.Set.Finite.Lattice
/-!
# Partitions based on membership of a sequence of sets
Let `f : ℕ → Set α` be a sequence of sets. For `n : ℕ`, we can form th... | (hs : s ∈ memPartition f n) :
memPartitionSet f n a = s ↔ a ∈ s := by
refine ⟨fun h ↦ h ▸ mem_memPartitionSet f n a, fun h ↦ ?_⟩
by_contra h_ne
have h_disj : Disjoint s (memPartitionSet f n a) :=
disjoint_memPartition f n hs (memPartitionSet_mem f n a) (Ne.symm h_ne)
refine absurd h_disj ?_
rw [no... | Mathlib/Data/Set/MemPartition.lean | 141 | 150 |
/-
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... | toProd
| Mathlib/Order/Interval/Basic.lean | 47 | 48 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.