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 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Damiano Testa, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mat... | have : degree (C (p.coeff 0)) < degree (divX p * X) :=
calc
| Mathlib/Algebra/Polynomial/Inductions.lean | 116 | 117 |
/-
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 | 1,738 | 1,738 | |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Math... | /-- The inversion of Ore fractions for a ring without zero divisors, satisfying `0⁻¹ = 0` and
`(r /ₒ r')⁻¹ = r' /ₒ r` for `r ≠ 0`. -/
@[irreducible]
protected def inv : R[R⁰⁻¹] → R[R⁰⁻¹] :=
liftExpand
(fun r s =>
if hr : r = (0 : R) then (0 : R[R⁰⁻¹])
else s /ₒ ⟨r, fun _ => eq_zero_of_ne_zero_of_mul_r... | Mathlib/RingTheory/OreLocalization/Ring.lean | 235 | 242 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... |
theorem stoppedProcess_eq (n : ℕ) : stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) +
∑ i ∈ Finset.range n, Set.indicator {ω | τ ω = i} (u i) := by
rw [stoppedProcess_eq'' n]
congr with i
rw [Finset.mem_Iio, Finset.mem_range]
theorem stoppedProcess_eq' (n : ℕ) : stoppedProcess u τ n = Set.indicator... | Mathlib/Probability/Process/Stopping.lean | 981 | 989 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Minimal
import Mathlib.Order.Zorn
import Mathlib.Topology.ContinuousOn
/-!
# Irreducibility in topological space... | simp [*]
theorem IsIrreducible.image (H : IsIrreducible s) (f : X → Y) (hf : ContinuousOn f s) :
IsIrreducible (f '' s) :=
| Mathlib/Topology/Irreducible.lean | 190 | 193 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | lemma iSup_induction {ι} (N : ι → LieSubmodule R L M) {motive : M → Prop} {x : M}
(hx : x ∈ ⨆ i, N i) (mem : ∀ i, ∀ y ∈ N i, motive y) (zero : motive 0)
(add : ∀ y z, motive y → motive z → motive (y + z)) : motive x := by
| Mathlib/Algebra/Lie/Submodule.lean | 448 | 450 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | ((Ico a b).image fun x => c - x) = Ico (c + 1 - b) (c + 1 - a) := by
ext x
simp_rw [mem_image, mem_Ico]
| Mathlib/Order/Interval/Finset/Nat.lean | 168 | 170 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.Measure... | refine or_iff_not_imp_right.mpr fun H => ?_; push_neg at H
refine hfi.ae_eq_of_forall_setIntegral_eq _ _ (integrable_const _) fun t ht ht' => ?_; clear ht'
simp only [const_apply, setIntegral_const]
by_cases h₀ : μ t = 0
| Mathlib/Analysis/Convex/Integral.lean | 220 | 223 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | · intro h' hs
exact h' _ hs rfl
theorem coe_deleteEdges_eq (s : Set (Sym2 V)) :
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,078 | 1,081 |
/-
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... | exact not_irreducible_pow (Fact.out : p.Prime).ne_one
instance instPerfectRingProd (S : Type*) [CommSemiring S] [ExpChar S p] [PerfectRing S p] :
PerfectRing (R × S) p where
| Mathlib/FieldTheory/Perfect.lean | 168 | 171 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | theorem mul_transvection_apply_of_ne (a b : n) (hb : b ≠ j) (c : R) (M : Matrix n n R) :
(M * transvection i j c) a b = M a b := by simp [transvection, Matrix.mul_add, hb]
| Mathlib/LinearAlgebra/Matrix/Transvection.lean | 125 | 127 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... | Mathlib/Order/Disjoint.lean | 293 | 293 | |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
/-!
# Extensionality lemmas for monoid and group structures
In this file we prove extensionality lemmas... | Mathlib/Algebra/Group/Ext.lean | 179 | 181 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 3,222 | 3,223 | |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | theorem IsLUB.unique (Ha : IsLUB s a) (Hb : IsLUB s b) : a = b :=
IsLeast.unique Ha Hb
| Mathlib/Order/Bounds/Basic.lean | 873 | 874 |
/-
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... | Mathlib/Data/Option/Basic.lean | 471 | 474 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
/-!
# The Jacobi Symbol
We define the Jacobi symbol and prove its main pro... | qrSign.symm hpo ha]
refine value_at p (rhs p) (fun q pq hq => ?_) ha
| Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 411 | 412 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 509 | 515 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,467 | 2,478 | |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,484 | 1,486 | |
/-
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.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.GDelta.UniformSpace
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathl... |
instance Empty.borelSpace : BorelSpace Empty :=
⟨borel_eq_top_of_discrete.symm⟩
instance Unit.borelSpace : BorelSpace Unit :=
⟨borel_eq_top_of_discrete.symm⟩
| Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 650 | 656 |
/-
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... |
section CharThree
variable [CharP R 3]
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 319 | 322 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,397 | 1,405 | |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | variable (R : Type*) [Semiring R] (r : R) (p q : R[X]) {S : Type*}
[NonAssocSemiring S] [Module R S] [Pow S ℕ] (x : S)
theorem smeval_C_mul : (C r * p).smeval x = r • p.smeval x := by
induction p using Polynomial.induction_on' with
| add p q ph qh => simp only [mul_add, smeval_add, ph, qh, smul_add]
| monomial... | Mathlib/Algebra/Polynomial/Smeval.lean | 180 | 187 |
/-
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.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Order.Mono... | alias ⟨MonovaryOn.sub_smul_sub_nonneg, _⟩ := monovaryOn_iff_forall_smul_nonneg
alias ⟨AntivaryOn.sub_smul_sub_nonpos, _⟩ := antivaryOn_iff_forall_smul_nonpos
alias ⟨Monovary.sub_smul_sub_nonneg, _⟩ := monovary_iff_forall_smul_nonneg
alias ⟨Antivary.sub_smul_sub_nonpos, _⟩ := antivary_iff_forall_smul_nonpos
alias ⟨Monov... | Mathlib/Algebra/Order/Monovary.lean | 405 | 409 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Jakob von Raumer
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Module.End
import Mathlib.CategoryTheory.Endomo... | section
| Mathlib/CategoryTheory/Preadditive/Basic.lean | 286 | 287 |
/-
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.Algebra.Group.Fin.Basic
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Tactic.Abel
/-!
# Circulant matrices
This file contains the defini... |
@[simp]
theorem circulant_zero (α n) [Zero α] [Sub n] : circulant 0 = (0 : Matrix n n α) :=
ext fun _ _ => rfl
| Mathlib/LinearAlgebra/Matrix/Circulant.lean | 95 | 98 |
/-
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... |
end CommRing
section IsDomain
variable {R : Type u} [CommRing R] [IsDomain R]
| Mathlib/FieldTheory/Separable.lean | 288 | 294 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | f.degree_erase_le _
theorem degree_eraseLead_lt (hf : f ≠ 0) : (eraseLead f).degree < f.degree :=
f.degree_erase_lt hf
theorem eraseLead_natDegree_le_aux : (eraseLead f).natDegree ≤ f.natDegree :=
natDegree_le_natDegree eraseLead_degree_le
theorem eraseLead_natDegree_lt (f0 : 2 ≤ #f.support) : (eraseLead f).na... | Mathlib/Algebra/Polynomial/EraseLead.lean | 182 | 190 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury, Scott Carnahan
-/
import Mathlib.LinearAlgebra.PerfectPairing.Basic
import Mathlib.LinearAlgebra.Reflection
/-!
# Root data and root systems
This file con... | lemma ne_zero' [NeZero (2 : R)] : (P.coroot i : N) ≠ 0 :=
P.flip.ne_zero i
lemma exists_ne_zero [Nonempty ι] [NeZero (2 : R)] : ∃ i, P.root i ≠ 0 := by
| Mathlib/LinearAlgebra/RootSystem/Defs.lean | 170 | 173 |
/-
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.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | @[reassoc (attr := simp)]
lemma map_ofInv_hom_id (s : Y ⟶ X) (hs : W s) (L : C ⥤ D) (hL : W.IsInvertedBy L) :
(ofInv s hs).map L hL ≫ L.map s = 𝟙 _ := by
letI := hL _ hs
simp [map]
| Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 153 | 157 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | rw [← ofFinsupp_neg, coeff, coeff, Finsupp.neg_apply]
| Mathlib/Algebra/Polynomial/Basic.lean | 1,103 | 1,104 |
/-
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.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... |
/-- Helper definition for the seminorms of the Schwartz space. -/
protected def seminormAux (k n : ℕ) (f : 𝓢(E, F)) : ℝ :=
sInf { c | 0 ≤ c ∧ ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ c }
theorem seminormAux_nonneg (k n : ℕ) (f : 𝓢(E, F)) : 0 ≤ f.seminormAux k n :=
le_csInf (bounds_nonempty k n f) fun _ ⟨hx, _⟩... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 194 | 200 |
/-
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.Combinatorics.SimpleGraph.Regularity.Chunk
import Mathlib.Combinatorics.SimpleGraph.Regularity.Energy
/-!
# Increment partitio... | rw [distinctPairs, ← add_sub_assoc, add_sub_right_comm]
split_ifs with h
· rw [add_zero]
exact edgeDensity_chunk_uniform hPα hPε _ _
· exact edgeDensity_chunk_not_uniform hPα hPε hε₁ (mem_offDiag.1 x.2).2.2 h
/-- The increment partition has energy greater than the original one by a known fixed amount. -/
t... | Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean | 126 | 135 |
/-
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.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | | succ n ih => rw [pow_succ A, mul_inv_rev, ← ih, ← pow_succ']
theorem pow_sub' (A : M) {m n : ℕ} (ha : IsUnit A.det) (h : n ≤ m) :
A ^ (m - n) = A ^ m * (A ^ n)⁻¹ := by
| Mathlib/LinearAlgebra/Matrix/ZPow.lean | 44 | 47 |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Bitraversable.Basic
/-!
# Bitraversable Lemmas
## Main definitions
* tfst - traverse on first functor argument
* tsnd - traverse on second func... | Mathlib/Control/Bitraversable/Lemmas.lean | 116 | 118 | |
/-
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... | exact ((Partrec.rfind hp).bind (hF.bind (sumCasesOn_right snd snd.to₂ none.to₂).to₂).to₂).of_eq
fun a => ext fun b => by simpa [p] using fix_aux f _ _
end Partrec
| Mathlib/Computability/Partrec.lean | 766 | 772 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.Category.CompHausLike.Limits
/-!
# Explicit limits and colimits
This file applies... | Mathlib/Topology/Category/Profinite/Limits.lean | 195 | 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.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | ### Algebraic replacement rules
| Mathlib/Order/Filter/NAry.lean | 160 | 160 |
/-
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.... |
/-- The tautological action by `Module.End R M` (aka `M →ₗ[R] M`) on `M`.
This generalizes `Function.End.applyMulAction`. -/
| Mathlib/Algebra/Module/LinearMap/End.lean | 185 | 189 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Order.RelClasses
/-!
# Euclidean domains
This file introduces Euclidean domains and provides the extende... | /-- `gcd a b` is a (non-unique) element such that `gcd a b ∣ a` `gcd a b ∣ b`, and for
any element `c` such that `c ∣ a` and `c ∣ b`, then `c ∣ gcd a b` -/
| Mathlib/Algebra/EuclideanDomain/Defs.lean | 165 | 166 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | val := ⟨x, y⟩
| Mathlib/NumberTheory/Pell.lean | 133 | 133 |
/-
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.HomotopyCofiber
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Algebra.Homology.QuasiIso
import Mathlib.CategoryTheory.Localiz... | lemma mem_quasiIso_iff {X Y : HomotopyCategory C c} (f : X ⟶ Y) :
quasiIso C c f ↔ ∀ (n : ι), IsIso ((homologyFunctor _ _ n).map f) := by
rfl
| Mathlib/Algebra/Homology/Localization.lean | 104 | 106 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x) :
ContDiffWithinAt 𝕜 n (fun x : E => (f x, g x)) s x := by
match n with
| ω =>
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 493 | 496 |
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Anthony DeRossi
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespac... | reduceOption (none :: l) = l.reduceOption := by simp only [reduceOption, filterMap, id]
| Mathlib/Data/List/ReduceOption.lean | 25 | 26 |
/-
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.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Typeclasses... | containing one of the points but not the other. -/
lemma dirac_ne_dirac_iff_exists_measurableSet {x y : α} :
Measure.dirac x ≠ Measure.dirac y ↔ ∃ A, MeasurableSet A ∧ x ∈ A ∧ y ∉ A := by
apply not_iff_not.mp
simp only [ne_eq, not_not, not_exists, not_and, dirac_eq_dirac_iff_forall_mem_iff_mem]
refine ⟨fun h ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 206 | 211 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | (since := "2025-04-10")]
structure LinearOrderedAddCommGroup (α : Type u) extends OrderedAddCommGroup α, LinearOrder α
| Mathlib/Algebra/Order/Group/Defs.lean | 82 | 84 |
/-
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... | p.order = Nat.find hp := by convert Nat.sInf_def hp
theorem order_eq_find' [DecidablePred fun n => p n ≠ 0] (hp : p ≠ 0) :
| Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 263 | 265 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | @[simp]
theorem vec2_dotProduct (v w : Fin 2 → α) : v ⬝ᵥ w = v 0 * w 0 + v 1 * w 1 :=
| Mathlib/Data/Matrix/Notation.lean | 510 | 511 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | theorem _root_.List.toFinset_filterMap [DecidableEq α] [DecidableEq β] (s : List α) :
(s.filterMap f).toFinset = s.toFinset.filterMap f f_inj := by
simp [← Finset.coe_inj]
| Mathlib/Data/Finset/Image.lean | 560 | 562 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,133 | 3,146 | |
/-
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 | 1,561 | 1,563 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Algebra.Group.Subsemigroup.Defs
/-!
# Centers of semigroups, as subsemigroups.
## Main definitions
* `S... | decidable_of_iff' _ Semigroup.mem_center_iff
end Semigroup
| Mathlib/GroupTheory/Subsemigroup/Center.lean | 66 | 68 |
/-
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.Multiset.Dedup
import Mathlib.Data.List.Infix
/-!
# Preparations for defining operations on `Finset`.
The operations here ignore multiplicities,... | ∀ h : ∀ p : { x // x ∈ s }, p.1 ∈ s,
(fun p : { x // x ∈ s } => ⟨p.val, h p⟩ : { x // x ∈ s } → { x // x ∈ s }) = id :=
fun _ => funext fun _ => Subtype.eq rfl
have : ∀ (t) (eq : s.ndinsert a = t), t.attach = ndinsert ⟨a, eq ▸ mem_ndinsert_self a s⟩
(s.attach.map fun p => ⟨p.1, eq ▸ mem_ndinsert_o... | Mathlib/Data/Multiset/FinsetOps.lean | 91 | 97 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... | ⟨hf.1, hf.2.absolutelyContinuous⟩
protected theorem comp {g : β → γ} {f : α → β} (hg : MeasurePreserving g μb μc)
| Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 87 | 89 |
/-
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.Logic.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... | @[to_additive "If a function is countably sub-additive then it is binary sub-additive"]
theorem rel_sup_mul [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R (m (⨆ i, s i)) (∏' i, m (s i))) (s₁ s₂ : α) :
R (m (s₁ ⊔ s₂)) (m s₁ * m s₂) := by
convert rel_iSup_tprod m m0 R... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 174 | 179 |
/-
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 | 1,204 | 1,207 | |
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Antidiag.Pi
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
im... | rw [sum_congr rfl this]
simp only [Nat.antidiagonal_eq_map, sum_map, Function.Embedding.coeFn_mk]
rw [(Commute.sum_right _ _ _ fun i hi ↦ hc (by simp) (by simp [hi])
(by simpa [eq_comm] using ne_of_mem_of_not_mem hi has)).add_pow]
simp only [ih (hc.mono (by simp)), sum_mul, mul_sum]
refine sum... | Mathlib/Data/Nat/Choose/Multinomial.lean | 231 | 267 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... | @[simp]
theorem mk_preimage_sigma (hi : i ∈ s) : Sigma.mk i ⁻¹' s.sigma t = t i :=
ext fun _ ↦ and_iff_right hi
| Mathlib/Data/Set/Sigma.lean | 179 | 181 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.LinearAl... | | smul₀ => aesop
| @smul₁ t n mem h =>
rcases h with ⟨n, hn, rfl⟩
exact ⟨t • n, by aesop, smul_comm _ _ _⟩
| add mem₁ mem₂ h₁ h₂ =>
rcases h₁ with ⟨m₁, h₁, rfl⟩
rcases h₂ with ⟨m₂, h₂, rfl⟩
exact ⟨m₁ + m₂, Submodule.add_mem _ h₁ h₂, by simp⟩
| zero => exact ⟨0, Submodule.z... | Mathlib/Algebra/Module/Submodule/Pointwise.lean | 473 | 484 |
/-
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.ContMDiff.Basic
/-!
## Smoothness of charts and local structomorphisms
We show that the model with corners,... | simp only [mfld_simps]
refine contDiffWithinAt_id.congr_of_eventuallyEq ?_ ?_
· exact Filter.eventuallyEq_of_mem self_mem_nhdsWithin fun x₂ => I.right_inv
simp_rw [Function.comp_apply, I.left_inv, Function.id_def]
theorem contMDiffOn_model_symm : ContMDiffOn 𝓘(𝕜, E) I n I.symm (range I) := by
rw [contMDiff... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 36 | 42 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... | apply Exists.intro w
cases lt_or_le (x' w) (a w) <;> simp_all
| Mathlib/Order/Interval/Set/Pi.lean | 296 | 298 |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... | cases p
· rfl
· simp only [Nat.succ_ne_zero, length_cons] at hzero
| Mathlib/Combinatorics/Quiver/Cast.lean | 130 | 133 |
/-
Copyright (c) 2020 Bhavik Mehta, Edward Ayers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Category.P... | J.Covers (S ⊓ R) f := by simpa [covers_iff] using And.intro hS hR
variable (C)
| Mathlib/CategoryTheory/Sites/Grothendieck.lean | 197 | 200 |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.... | section OrthogonalGroup
variable (n) (β : Type v) [CommRing β]
| Mathlib/LinearAlgebra/UnitaryGroup.lean | 190 | 192 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | to `IsStronglyCoatomic`/`IsStronglyAtomic` orders. -/
end Subset
| Mathlib/Order/Minimal.lean | 347 | 349 |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Tactic.Group
import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere positive sets in measure spaces
... |
/-- An open set is everywhere positive for a measure which is positive on open sets. -/
lemma _root_.IsOpen.isEverywherePos [IsOpenPosMeasure μ] (hs : IsOpen s) : IsEverywherePos μ s := by
intro x xs n hn
rcases mem_nhdsWithin.1 hn with ⟨u, u_open, xu, hu⟩
apply lt_of_lt_of_le _ (measure_mono hu)
| Mathlib/MeasureTheory/Measure/EverywherePos.lean | 192 | 197 |
/-
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... | variable [TopologicalSpace Ω]
theorem testAgainstNN_eq_mass_mul (f : Ω →ᵇ ℝ≥0) :
μ.testAgainstNN f = μ.mass * μ.normalize.toFiniteMeasure.testAgainstNN f := by
| Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 437 | 440 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Normed.Module.Convex
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neum... |
section
variable [T1Space E]
| Mathlib/Analysis/Convex/GaugeRescale.lean | 58 | 61 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.Algebra.Ring.Action.Group
/-!
# Isomorphisms of `R`-algebras
This file defines bundled isomorphisms of `R`-... | (AlgEquiv.ofBijective f hf) a = f a :=
rfl
/-- Forgetting the multiplicative structures, an equivalence of algebras is a linear equivalence. -/
@[simps apply]
| Mathlib/Algebra/Algebra/Equiv.lean | 482 | 486 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... |
/-!
## Square roots of reals
-/
namespace Real
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 967 | 973 |
/-
Copyright (c) 2016 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.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | exact eq_of_heq (this a)
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
lemma funext_iff_of_subsingleton [Subsingleton α] {g : α → β} (x y : α) :
f x = g y ↔ f = g := by
refine ⟨fun h ↦ funext fun z ↦ ?_, fun h ↦ ?_⟩
· rwa [Subsingleton.el... | Mathlib/Logic/Function/Basic.lean | 57 | 65 |
/-
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... | · suffices h_balanceable : _ by
constructor
| Mathlib/Data/Ordmap/Ordset.lean | 595 | 596 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... |
theorem ContDiffOn.comp_contDiff {s : Set F} {g : F → G} {f : E → F} (hg : ContDiffOn 𝕜 n g s)
(hf : ContDiff 𝕜 n f) (hs : ∀ x, f x ∈ s) : ContDiff 𝕜 n (g ∘ f) := by
rw [← contDiffOn_univ] at *
exact hg.comp hf fun x _ => hs x
theorem ContDiffOn.image_comp_contDiff {s : Set E} {g : F → G} {f : E → F}
(... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 632 | 660 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
/-!
# Racks and Quandles
This file defines racks and quandles, algebraic structu... | theorem op_invAct_op_eq {x y : R} : op x ◃⁻¹ op y = op (x ◃ y) :=
rfl
| Mathlib/Algebra/Quandle.lean | 251 | 253 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Emily Riehl
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Functor.TwoSquare
import Mathlib.CategoryTheory.HomCongr
import Mathlib.Tactic.... | have vcomp :=
mateEquiv_vcomp (adj₁.comp adj₂) (adj₃.comp adj₄) (adj₅.comp adj₆) (α ≫ᵥ β) (γ ≫ᵥ δ)
simp only [mateEquiv_hcomp] at vcomp
| Mathlib/CategoryTheory/Adjunction/Mates.lean | 223 | 225 |
/-
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 | 1,366 | 1,371 | |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Tactic.NormNum.Basic
/-!
## `norm_num` plugin for `^`.
-/
assert_not_exists RelIso
namespace Ma... | (pb : IsInt b (Int.negOfNat nb)) (pe' : IsInt (a ^ nb)⁻¹ (Int.negOfNat ne)) :
IsInt (a ^ b) (Int.negOfNat ne) := by
rwa [pb.out, Int.cast_negOfNat, zpow_neg, zpow_natCast]
theorem isRat_zpow_pos {α : Type*} [DivisionRing α] {a : α} {b : ℤ} {nb : ℕ}
| Mathlib/Tactic/NormNum/Pow.lean | 216 | 220 |
/-
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.... | @[simp]
theorem realize_liftAt {n n' m : ℕ} {t : L.Term (α ⊕ (Fin n))} {v : α ⊕ (Fin (n + n')) → M} :
(t.liftAt n' m).realize v =
t.realize (v ∘ Sum.map id fun i : Fin _ =>
if ↑i < m then Fin.castAdd n' i else Fin.addNat i n') :=
| Mathlib/ModelTheory/Semantics.lean | 88 | 92 |
/-
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... | simp [addCases, this]
theorem append_snoc {α : Sort*} (as : Fin n → α) (bs : Fin m → α) (b : α) :
Fin.append as (snoc bs b) = snoc (Fin.append as bs) b := by
funext i
rcases i with ⟨i, isLt⟩
simp only [append, addCases, castLT, cast_mk, subNat_mk, natAdd_mk, cast, snoc.eq_1,
cast_eq, eq_rec_constan... | Mathlib/Data/Fin/Tuple/Basic.lean | 670 | 686 |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | theorem yn_ge_n : ∀ n, n ≤ yn a1 n
| 0 => Nat.zero_le _
| n + 1 =>
| Mathlib/NumberTheory/PellMatiyasevic.lean | 359 | 361 |
/-
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.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.FunctorCategory.EpiMono
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.Categ... | def Sheaf.imageι {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf.image f ⟶ F' :=
⟨Subpresheaf.ι _⟩
@[reassoc (attr := simp)]
theorem Sheaf.toImage_ι {F F' : Sheaf J (Type w)} (f : F ⟶ F') :
toImage f ≫ imageι f = f := by
ext1
simp [Subpresheaf.toRangeSheafify]
instance {F F' : Sheaf J (Type w)} (f : F ⟶ F')... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 216 | 255 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | rw [← div_eq_iff (hne p), div_eq_mul_inv, sum_mul]
rw [PowerSeries.ext_iff] at this
simpa using this p
-- the power series `exp ℚ - 1` is non-zero, a fact we need in order to use `mul_right_inj'`
have hexp : exp ℚ - 1 ≠ 0 := by
simp only [exp, PowerSeries.ext_iff, Ne, not_forall]
u... | Mathlib/NumberTheory/Bernoulli.lean | 305 | 369 |
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | theorem infDist_le_infDist_add_dist : infDist x s ≤ infDist y s + dist x y := by
rw [infDist, infDist, dist_edist]
refine ENNReal.toReal_le_add' infEdist_le_infEdist_add_edist ?_ (flip absurd (edist_ne_top _ _))
simp only [infEdist_eq_top_iff, imp_self]
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 488 | 491 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | rw [@eq_comm _ _ φ, @eq_comm _ _ (ComplexEmbedding.conjugate φ), ← mk_eq_iff, mk_embedding]
@[simp]
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 367 | 369 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Fintype.Order
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Tactic.Linarith
/-!
# Acyclic grap... |
theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q := by
| Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 83 | 85 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... |
@[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 132 | 133 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | rintro rfl
norm_num [← Nat.not_even_iff_odd] at h2
rw [reciprocalFactors, dif_neg h0, dif_neg h1, if_neg (Nat.not_even_iff_odd.2 h2)]
/-- A finite product of Dihedral groups. -/
abbrev Product (l : List ℕ) : Type :=
| Mathlib/GroupTheory/CommutingProbability.lean | 171 | 176 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | · exact ⟨(hf.mono hVs').isOpen_inter_preimage hV hU.1,
subset_inter hsV (hst.mono Subset.rfl hU.2)⟩
· intro h
rw [← mem_preimage]
exact mem_of_mem_inter_right h
| Mathlib/Topology/ContinuousOn.lean | 1,370 | 1,374 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | | Sum.inl a => (some a, Sum.inl a)
| Sum.inr b =>
| Mathlib/Data/Seq/Computation.lean | 175 | 176 |
/-
Copyright (c) 2022 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Analysis.Convex.Cone.Closure
import Mathlib.Analysis.InnerProductSpace.Adjoint
/-!
# Proper cones
We define a *proper cone* as a closed, pointed cone. ... | @[ext]
theorem ext {S T : ProperCone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=
| Mathlib/Analysis/Convex/Cone/Proper.lean | 66 | 67 |
/-
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.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | rw [genWeightSpace, LieSubmodule.mem_iInf]
intro y
replace hx : x ∈ genWeightSpaceOf L (χ₁ y) y := by
rw [rootSpace, genWeightSpace, LieSubmodule.mem_iInf] at hx; exact hx y
replace hm : m ∈ genWeightSpaceOf M (χ₂ y) y := by
rw [genWeightSpace, LieSubmodule.mem_iInf] at hm; exact hm y
exact lie_mem_ma... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 61 | 69 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | @[simp]
theorem mem_ker {m : M} : m ∈ f.ker ↔ f m = 0 :=
Iff.rfl
| Mathlib/Algebra/Lie/Submodule.lean | 948 | 950 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | ∀ w : InfinitePlace K, w a < max B 1 := by
classical
have : minkowskiBound K ↑1 < volume (convexBodyLT K (fun w ↦ if w = w₀ then B else 1)) := by
rw [convexBodyLT_volume, ← Finset.prod_erase_mul _ _ (Finset.mem_univ w₀)]
simp_rw [ite_pow, one_pow]
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 532 | 536 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | @[simp]
theorem sublistForall₂_map_right_iff {f : γ → β} {l₁ : List α} {l₂ : List γ} :
SublistForall₂ R l₁ (map f l₂) ↔ SublistForall₂ (fun a c => R a (f c)) l₁ l₂ := by
simp only [sublistForall₂_iff]
constructor
· rintro ⟨l1, h1, h2⟩
obtain ⟨l', hl1, rfl⟩ := sublist_map_iff.mp h2
use l'
simpa [hl... | Mathlib/Data/List/Forall2.lean | 332 | 348 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,749 | 1,750 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,096 | 1,100 | |
/-
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.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 286 | 289 | |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois.Basic
/-!
# Cyclotomic extens... | adjoin_roots := fun ⟨x, hx⟩ => by
refine
adjoin_induction
(hx := hx) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ _ _ hb₁ hb₂ => ?_)
(fun b₁ b₂ _ _ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine subset_adjoin ?_
simp only [mem_singleton_iff, exists_eq_l... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 387 | 399 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | @[to_additive (attr := simp) AddMonoid.addOrderOf_eq_one_iff]
theorem orderOf_eq_one_iff : orderOf x = 1 ↔ x = 1 := by
| Mathlib/GroupTheory/OrderOfElement.lean | 227 | 228 |
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