Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | theorem cycleRange_last (n : ℕ) : cycleRange (last n) = finRotate (n + 1) := by
ext i
| Mathlib/GroupTheory/Perm/Fin.lean | 205 | 206 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,104 | 1,108 | |
/-
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.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | @[simp]
theorem modByMonic_zero (p : R[X]) : p %ₘ 0 = p :=
letI := Classical.decEq R
if h : Monic (0 : R[X]) then by
haveI := monic_zero_iff_subsingleton.mp h
simp [eq_iff_true_of_subsingleton]
else by unfold modByMonic divModByMonicAux; rw [dif_neg h]
| Mathlib/Algebra/Polynomial/Div.lean | 186 | 192 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.MappingCone
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift
import Mathlib.CategoryTheory.Triangulated.Functor... | @[simps! obj₁ obj₂ obj₃ mor₁ mor₂]
noncomputable def triangle : Triangle (CochainComplex C ℤ) :=
Triangle.mk φ (inr φ) (Cocycle.homOf ((-fst φ).rightShift 1 0 (zero_add 1)))
@[reassoc (attr := simp)]
| Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean | 49 | 53 |
/-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Types.Limits
impo... | ((binaryProductIso F G).inv).app a ⟨x, y⟩
@[simp]
lemma prodMk_fst {a : C} (x : F.obj a) (y : G.obj a) :
| Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean | 125 | 128 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | { Matrix.linftyOpNonUnitalSemiNormedRing with
eq_of_dist_eq_zero := eq_of_dist_eq_zero }
| Mathlib/Analysis/Matrix.lean | 366 | 368 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... |
lemma comp_eq_zero_iff [Semiring R] [NoZeroDivisors R] {p q : R[X]} :
p.comp q = 0 ↔ p = 0 ∨ p.eval (q.coeff 0) = 0 ∧ q = C (q.coeff 0) := by
refine ⟨fun h ↦ ?_, Or.rec (fun h ↦ by simp [h]) fun h ↦ by rw [h.2, comp_C, h.1, C_0]⟩
have key : p.natDegree = 0 ∨ q.natDegree = 0 := by
rw [← mul_eq_zero, ← natDe... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 380 | 392 |
/-
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.GroupWithZero.Action.Pi
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Pi
import Mathlib.Data.Finsupp.Order
/-!
# Flooring,... |
lemma floorDiv_le_ceilDiv : b ⌊/⌋ a ≤ b ⌈/⌉ a := by
obtain ha | ha := le_or_lt a 0
· simp [ha]
| Mathlib/Algebra/Order/Floor/Div.lean | 123 | 126 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | Mathlib/Data/Bool/Basic.lean | 236 | 236 | |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | rw [← ha, mul_assoc, mul_assoc, ← mul_assoc a z⁻¹ a⁻¹]
exact mul_mem hz (Normal.conj_mem (lowerCentralSeries_normal n) z⁻¹ (inv_mem hz) a)
/-- The lower central series of a group is a descending central series. -/
theorem lowerCentralSeries_isDescendingCentralSeries :
| Mathlib/GroupTheory/Nilpotent.lean | 324 | 329 |
/-
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 `... | Mathlib/Data/Finset/Sups.lean | 725 | 725 | |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`
Useful for defining group cohomology."]
theorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)
(j : Fin (n + 1)) (k : Fin n) :
| Mathlib/Algebra/BigOperators/Fin.lean | 251 | 254 |
/-
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... | Mathlib/Algebra/Order/Group/Defs.lean | 1,077 | 1,080 | |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | · refine le_sInf ?_
intro μ hμ t
exact le_trans (boundedBy_le t) (iInf₂_le μ hμ)
theorem iSup_sInfGen_nonempty {m : Set (OuterMeasure α)} (h : m.Nonempty) (t : Set α) :
⨆ _ : t.Nonempty, sInfGen m t = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), μ t := by
rcases t.eq_empty_or_nonempty with (rfl | ht)
· simp [... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 346 | 353 |
/-
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... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 475 | 478 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | end Prod
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 171 | 172 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Nat.Factors
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
impo... | intro n h
obtain hdvd|⟨p, hpprime, hdvd, hpodd⟩ := Nat.four_dvd_or_exists_odd_prime_and_dvd_of_two_lt h <;>
apply FermatLastTheoremWith.mono hdvd
· exact fermatLastTheoremFour
· exact hprimes p hpprime hpodd
| Mathlib/NumberTheory/FLT/Four.lean | 309 | 313 |
/-
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... |
@[to_additive Right.nsmul_nonpos]
theorem Right.pow_le_one_of_le (hx : x ≤ 1) {n : ℕ} : x ^ n ≤ 1 :=
Right.one_le_pow_of_le (M := Mᵒᵈ) hx
| Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 103 | 107 |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | ⟨fun h => h.of_map fun _ => hg.1, fun h => h.map <| hg.2 rfl⟩
protected theorem pair {g : α →ₛ γ} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ) :
(pair f g).FinMeasSupp μ :=
calc
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,086 | 1,090 |
/-
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... | simp_rw [one_div]
| Mathlib/Data/Complex/Exponential.lean | 351 | 351 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | theorem inv_single {R} [DivisionRing R] {a b : R}
(_ : (a : R)⁻¹ = b) : (a + 0)⁻¹ = b + 0 := by simp [*]
| Mathlib/Tactic/Ring/Basic.lean | 968 | 969 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... |
section GroupWithZero
variable {G₀ : Type*} [GroupWithZero G₀] {x : G₀}
/-- Canonical isomorphism between the non-zero-divisors and units of a group with zero. -/
@[simps]
noncomputable def nonZeroDivisorsEquivUnits : G₀⁰ ≃* G₀ˣ where
toFun u := .mk0 _ <| mem_nonZeroDivisors_iff_ne_zero.1 u.2
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 258 | 265 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | | prec _ _ hf hg =>
simpa using
prec' (unpair₂ head) (hf.comp₁ _ (unpair₁ head))
(hg.comp₁ _ <|
natPair.comp₂ _ (unpair₁ <| tail <| tail head) (natPair.comp₂ _ head (tail head)))
| Mathlib/Computability/Primrec.lean | 1,403 | 1,407 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
/-!
# Degreewise split exact sequences of cochain complexes
The main result of this file is the lemma
`Homoto... | degreewise split short exact sequence of cochain complexes. -/
noncomputable def trianglehRotateIsoTrianglehOfDegreewiseSplit :
(triangleh φ).rotate ≅
trianglehOfDegreewiseSplit _ (triangleRotateShortComplexSplitting φ) :=
(HomotopyCategory.quotient _ _).mapTriangleRotateIso.app _ ≪≫
(HomotopyCategory.q... | Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean | 210 | 225 |
/-
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.QuadraticForm.IsometryEquiv
import Mathlib.Algebra.Category.ModuleCat.Basic
/-!
# The category of quadratic modules
-/
open CategoryTheory
u... | instance category : Category (QuadraticModuleCat.{v} R) where
Hom M N := Hom M N
id M := ⟨Isometry.id M.form⟩
| Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean | 50 | 52 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Anatole Dedecker
-/
import Mathlib.Topology.Separation.Regular
/-!
# Extending a function from a subset
The main definition of this file is `extendFrom A f` where `... | intro V' V'_in V'_closed
obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ V ∩ A ⊆ f ⁻¹' V' := by
have := tendsto_extendFrom (hf x x_in)
rcases (nhdsWithin_basis_open x A).tendsto_left_iff.mp this V' V'_in with ⟨V, ⟨hxV, V_op⟩, hV⟩
exact ⟨V, IsOpen.mem_nhds V_op hxV, V_op, hV⟩
suffices ∀ y ∈ V ∩ B, ... | Mathlib/Topology/ExtendFrom.lean | 63 | 81 |
/-
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... |
The two directions are `YoungDiagram.rowLens` (defined above) and `YoungDiagram.ofRowLens`.
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 387 | 388 |
/-
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.Floor.Div
import Mathlib.Data.Nat.Factorization.Defs
/-!
# Roots of natural numbers, rounded up and down
This file defines the flooring and... | @[simp] lemma floorRoot_pow_self (hn : n ≠ 0) (a : ℕ) : floorRoot n (a ^ n) = a := by
simp [floorRoot_def, pos_iff_ne_zero.2, hn]; split_ifs <;> simp [*]
| Mathlib/Data/Nat/Factorization/Root.lean | 64 | 65 |
/-
Copyright (c) 2021 Alex Kontorovich, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Alge... | Mathlib/Topology/Algebra/ConstMulAction.lean | 567 | 571 | |
/-
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... | let t := yn b1 k
have sx : s ≡ x [MOD u] := (xy_modEq_of_modEq b1 a1 ba k).left
have tk : t ≡ k [MOD 4 * y] :=
have : 4 * y ∣ b - 1 :=
Int.natCast_dvd_natCast.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd
(yn_modEq_a_sub_one _ _).of_dvd this
⟨ky,
... | Mathlib/NumberTheory/PellMatiyasevic.lean | 776 | 797 |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra... | theorem divp_sub_divp_same (a b : α) (u : αˣ) : a /ₚ u - b /ₚ u = (a - b) /ₚ u := by
rw [sub_eq_add_neg, sub_eq_add_neg, neg_divp, divp_add_divp_same]
| Mathlib/Algebra/Ring/Units.lean | 61 | 62 |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Dynamics.PeriodicPts.Defs
import Mathlib.GroupTheory.G... | @[to_additive "If `(fixedBy α g)ᶜ ⊆ s`, then `g` cannot move a point of `s` outside of `s`."]
theorem set_mem_fixedBy_of_movedBy_subset {s : Set α} {g : G} (s_subset : (fixedBy α g)ᶜ ⊆ s) :
s ∈ fixedBy (Set α) g := by
rw [← fixedBy_inv]
ext a
rw [Set.mem_inv_smul_set_iff]
by_cases a ∈ fixedBy α g
case pos... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 161 | 172 |
/-
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.MeasureTheory.Covering.Differentiation
/-!
# Besicovitch covering theorems
The topological Besicovitch covering theorem ensures that, in a nice... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 1,131 | 1,139 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.Basis.Defs
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.RingTheory.Ideal.Span
/-!
# The b... | simp only [basisSpanSingleton, Basis.map_apply, LinearEquiv.trans_apply,
Submodule.restrictScalarsEquiv_apply, LinearEquiv.ofInjective_apply, LinearEquiv.coe_ofEq_apply,
LinearEquiv.restrictScalars_apply, LinearMap.mulLeft_apply, LinearMap.mul_apply']
@[simp]
| Mathlib/RingTheory/Ideal/Basis.lean | 35 | 39 |
/-
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.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | lemma homologyι_comp_asIsoHomologyι_inv (hg : S.g = 0) [S.HasHomology] :
S.homologyι ≫ (S.asIsoHomologyι hg).inv = 𝟙 _ := (S.asIsoHomologyι hg).hom_inv_id
lemma mono_homologyMap_of_mono_opcyclesMap'
[S₁.HasHomology] [S₂.HasHomology] (h : Mono (opcyclesMap φ)) :
Mono (homologyMap φ) := by
have : Mono (h... | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 1,146 | 1,152 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 447 | 448 |
/-
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... | /-- There is exactly one filter on an empty type. -/
instance unique [IsEmpty α] : Unique (Filter α) where
default := ⊥
uniq := filter_eq_bot_of_isEmpty
theorem NeBot.nonempty (f : Filter α) [hf : f.NeBot] : Nonempty α :=
not_isEmpty_iff.mp fun _ ↦ hf.ne (Subsingleton.elim _ _)
/-- There are only two filters on... | Mathlib/Order/Filter/Basic.lean | 376 | 392 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... |
theorem opNorm_pi {ι' : Type v'} [Fintype ι'] {E' : ι' → Type wE'}
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 545 | 546 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | _ = l.rotate' n := by rw [rotate'_rotate', length_rotate', Nat.mod_add_div]
theorem rotate_eq_rotate' (l : List α) (n : ℕ) : l.rotate n = l.rotate' n :=
if h : l.length = 0 then by simp_all [length_eq_zero_iff]
else by
rw [← rotate'_mod,
rotate'_eq_drop_append_take (le_of_lt (Nat.mod_lt _ (Nat.pos_... | Mathlib/Data/List/Rotate.lean | 93 | 100 |
/-
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
/... |
/-- Let `p` be a seminorm on a vector space over a `NormedField`.
If there is a scalar `c` with `‖c‖>1`, then any `x` such that `p x ≠ 0` can be
moved by scalar multiplication to any `p`-shell of width `‖c‖`. Also recap information on the
value of `p` on the rescaling element that shows up in applications. -/
lemma re... | Mathlib/Analysis/Seminorm.lean | 1,186 | 1,193 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousMap... | _ = |bernsteinApproximation n f x - f x * 1| := by rw [mul_one]
_ = |bernsteinApproximation n f x - f x * ∑ k : Fin (n + 1), bernstein n k x| := by
rw [bernstein.probability]
_ = |∑ k : Fin (n + 1), (f k/ₙ - f x) * bernstein n k x| := by
simp [bernsteinApproximation, Finset.mul_sum, sub_mul]
... | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 231 | 300 |
/-
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... | pow_one, pow_card_eq_one', one_zpow, one_mul, eq_comm] at key
| Mathlib/GroupTheory/OrderOfElement.lean | 955 | 956 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.ZMod
import Mathlib.Data.Nat.Multiplicity
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.... | Mathlib/RingTheory/WittVector/Frobenius.lean | 315 | 316 | |
/-
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 | 2,058 | 2,059 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Violeta Hernández Palacios
-/
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Regular
import Mathlib.SetTheory.Cardinal.Contin... | · intro f H
choose I hI using fun n => (H n).1
simp_rw [exists_prop] at hI
refine ⟨_, Ordinal.lsub_lt_ord_lift ?_ fun n => (hI n).1,
iUnion_mem_generateMeasurableRec fun n => ⟨_, Ordinal.lt_lsub I n, (hI n).2⟩⟩
rw [mk_nat, lift_aleph0, isRegular_aleph_one.cof_omega_eq]
exact aleph0_lt_aleph_... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 |
/-
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... | Mathlib/Data/Set/Basic.lean | 1,960 | 1,965 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 805 | 807 | |
/-
Copyright (c) 2024 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
/-!
# Fixed-point-free automorphisms
This file defines fixed-point-free automorphisms and proves some basic properties.... | have key : φ g = g := by
rw [hφ.coe_eq_inv_of_involutive h2, inv_eq_iff_mul_eq_one, ← sq, ← hg, pow_orderOf_eq_one]
rw [hφ g key, orderOf_one] at hg
contradiction
theorem odd_card_of_involutive (hφ : FixedPointFree φ) (h2 : Function.Involutive φ) :
| Mathlib/GroupTheory/FixedPointFree.lean | 76 | 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, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,501 | 1,502 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
import Mathlib.CategoryTheory.Triangulated.Triangulated
import Mathlib.CategoryTheory.ComposableArrows
/-! The... | (mappingConeCompHomotopyEquiv f g).hom ≫
(triangle (mappingConeCompTriangle f g).mor₁).mor₃ =
(mappingConeCompTriangle f g).mor₃ := by
ext n
simp [map, mappingConeCompHomotopyEquiv, MappingConeCompHomotopyEquiv.hom,
| Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean | 147 | 151 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | (h : ∀ i ∈ s, sign (f i) = t) : sign (∑ i ∈ s, f i) = t := by
cases t
| Mathlib/Data/Sign.lean | 455 | 456 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | termination_by (a, b)
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 511 | 512 |
/-
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 | 744 | 744 | |
/-
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... | theorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by
rw [mul_comm]
exact mul_left_not_lt b h
| Mathlib/Algebra/EuclideanDomain/Defs.lean | 131 | 133 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
/-!
# Split a box along one or more hyperplanes
## Main definitions
A hyperplane `{x : ι → ℝ | x i = a}` spli... | open scoped Classical in
/-- The partition of `I : Box ι` into the boxes `I ∩ {y | y ≤ x i}` and `I ∩ {y | x i < y}`.
One of these boxes can be empty, then this partition is just the single-box partition `⊤`. -/
def split (I : Box ι) (i : ι) (x : ℝ) : Prepartition I :=
ofWithBot {I.splitLower i x, I.splitUpper i x}
... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 147 | 153 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... |
@[simp]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 433 | 434 |
/-
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.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 343 | 347 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | structure LinearOrderedRing (R : Type u) extends StrictOrderedRing R, LinearOrder R
set_option linter.deprecated false in
/-- A `LinearOrderedCommRing` is a commutative ring with a linear order such that addition is
monotone and multiplication by a positive number is strictly monotone. -/
@[deprecated "Use `[CommRing ... | Mathlib/Algebra/Order/Ring/Defs.lean | 338 | 344 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Combinatorics.Additive.FreimanHom
/-!
# Corners
This file defines corners, namely triples of the form `(x, y), (x, y + d), (x + d, y)`, and the
property ... | · exact hf.add_eq_add hx₁ hy₂ hx₂ hy₁
lemma isCornerFree_image (hf : IsAddFreimanIso 2 s t f) (hAs : A ⊆ s ×ˢ s) :
IsCornerFree (Prod.map f f '' A) ↔ IsCornerFree A := by
have : Prod.map f f '' A ⊆ t ×ˢ t :=
((hf.bijOn.mapsTo.prodMap hf.bijOn.mapsTo).mono hAs Subset.rfl).image_subset
rw [isCornerFree_iff... | Mathlib/Combinatorics/Additive/Corner/Defs.lean | 92 | 98 |
/-
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.Probability.Kernel.Composition.IntegralCompProd
import Mathlib.Probability.Kernel.Disintegration.StandardBorel
/-!
# Lebesgue and Bochner integrals of con... |
theorem Integrable.condKernel_ae {f : α × Ω → F} (hf_int : Integrable f ρ) :
∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a) := by
have hf_ae : AEStronglyMeasurable f ρ := hf_int.1
rw [← hf_ae.ae_integrable_condKernel_iff] at hf_int
exact hf_int.1
| Mathlib/Probability/Kernel/Disintegration/Integral.lean | 240 | 245 |
/-
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.Algebra.Order.Group.Finset
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Sym.Basic
/-!
# Equivalence between `Multiset` and `ℕ`-valued finitel... | Finsupp.toMultiset f = s ↔ f = Multiset.toFinsupp s :=
Multiset.toFinsupp.symm_apply_eq
| Mathlib/Data/Finsupp/Multiset.lean | 186 | 188 |
/-
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.Data.Finsupp.Lex
import Mathlib.Algebra.MvPolynomial.Degrees
/-!
# Variables of polynomials
This file establishes man... | @[simp]
theorem vars_one : (1 : MvPolynomial σ R).vars = ∅ :=
vars_C
theorem vars_pow (φ : MvPolynomial σ R) (n : ℕ) : (φ ^ n).vars ⊆ φ.vars := by
| Mathlib/Algebra/MvPolynomial/Variables.lean | 115 | 119 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.OfNat
/-!
# Cast of natural numbers
This file defines the *can... | apply congrArg
decide
| Mathlib/Data/Nat/Cast/Defs.lean | 202 | 203 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Tactic.AdaptationNote
/-!
# Inversion in an affine sp... | works for `x = c`. -/
theorem dist_inversion_center (c x : P) (R : ℝ) : dist (inversion c R x) c = R ^ 2 / dist x c := by
| Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 89 | 90 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... | apply isIntegral_trace
rw [IsIntegralClosure.isIntegral_iff (A := B)]
have ⟨z, _, hz⟩ := hIJ (FractionalIdeal.mul_mem_mul hy hx)
rw [mul_comm] at hz
exact ⟨z, hz⟩
lemma one_le_dual_one :
1 ≤ dual A K (1 : FractionalIdeal B⁰ L) :=
| Mathlib/RingTheory/DedekindDomain/Different.lean | 303 | 310 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Set.Accumulate
import Mathlib.Data.Set.Pairwise.Lattice
import Mathlib.MeasureTheory.PiSystem
/-!... | (hI : ↑I ⊆ C) :
Disjoint (⋃₀ (I : Set (Set α))) (⋃₀ hC.disjointOfDiffUnion hs hI) := by
rw [← hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion]; exact Set.disjoint_sdiff_right
lemma disjoint_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
Disjoint I (hC.disjointOfDiffUnion hs hI) := b... | Mathlib/MeasureTheory/SetSemiring.lean | 269 | 274 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | rw [← not_subsingleton_iff_nontrivial, subsingleton_iff]
-- todo: this can be made a `Unique` instance.
instance subsingleton_units : Subsingleton (ZMod 2)ˣ :=
⟨by decide⟩
| Mathlib/Data/ZMod/Basic.lean | 897 | 902 |
/-
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... | refine ⟨n, hn₁, fun m mn km => ?_⟩
rcases km.lt_or_eq_dec with km | km
· exact hn₂ _ mn km
| Mathlib/Computability/Partrec.lean | 752 | 754 |
/-
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.Ordering.Basic
import Mathlib.Order.Synonym
/-!
# Comparison
This file provides basic results about orderings and comparison in linear orders.
... | Mathlib/Order/Compare.lean | 241 | 241 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,366 | 1,368 | |
/-
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, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | exact (H1.of_mul_left_right.isUnit_of_dvd hb).mul_right_dvd.mpr ha
| Mathlib/Algebra/Divisibility/Units.lean | 208 | 209 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... |
theorem reflTransGen_iff_eq_or_transGen : ReflTransGen r a b ↔ b = a ∨ TransGen r a b := by
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· cases h with
| Mathlib/Logic/Relation.lean | 519 | 522 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
/-!
# Uniqueness of the conditional expect... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | 218 | 226 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | rwa [← LinearPMap.graph_map_snd_eq_range, toLinearPMap_graph_eq]
end SubmoduleToLinearPMap
end Submodule
| Mathlib/LinearAlgebra/LinearPMap.lean | 951 | 955 |
/-
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 | 1,038 | 1,042 | |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Al... | simp only [rTensor_apply_tmul_apply, coeff_monomial, ite_tmul]
lemma rTensor_apply_X_tmul (s : σ) (n : N) (d : σ →₀ ℕ) :
| Mathlib/RingTheory/TensorProduct/MvPolynomial.lean | 79 | 81 |
/-
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.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on mul... | theorem coeff_rename_ne_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : (rename f φ).coeff d ≠ 0) : ∃ u : σ →₀ ℕ, u.mapDomain f = d ∧ φ.coeff u ≠ 0 := by
contrapose! h
apply coeff_rename_eq_zero _ _ _ h
@[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constant... | Mathlib/Algebra/MvPolynomial/Rename.lean | 311 | 321 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... |
/-- Reducible form of `congrArg_mpr_hom_right` -/
| Mathlib/CategoryTheory/EqToHom.lean | 169 | 170 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | eq_one_or_norm_pos a
@[to_additive (attr := simp) nnnorm_eq_zero]
theorem nnnorm_eq_zero' : ‖a‖₊ = 0 ↔ a = 1 := by
| Mathlib/Analysis/Normed/Group/Basic.lean | 1,279 | 1,282 |
/-
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 | 830 | 830 | |
/-
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... | (hf : ∀ y, x ≤ y ∧ y ≤ x → x ≤ f y ∧ f y ≤ x) {t} (h : Valid t) : Valid (insertWith f x t) :=
(insertWith.valid_aux _ _ hf h ⟨⟩ ⟨⟩).1
| Mathlib/Data/Ordmap/Ordset.lean | 513 | 515 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set α) :
⋃ i, (if h : p i then f i h else g i h) = (⋃ (i) (h : p i), f i h) ∪ ⋃ (i) (h : ¬p i), g i h :=
iSup_dite _ _ _
| Mathlib/Data/Set/Lattice.lean | 398 | 400 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies, David Loeffler
-/
import Mathlib.Order.PartialSups
import Mathlib.Order.Interval.Finset.Fin
/-!
# Making a sequence disjoint
This file defines the way t... | section PartialOrder -- the index type is a partial order
| Mathlib/Order/Disjointed.lean | 105 | 106 |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Pointwise.Stabilizer
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTh... | section SMul
variable (G : Type*) {X : Type*} [SMul G X] {B : Set X} {a : X}
| Mathlib/GroupTheory/GroupAction/Blocks.lean | 85 | 87 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Mates
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 190 | 193 | |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_left {I a₀ b : ℝ}
(hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
(h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) :=
integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi ha tendsto_const_nhds h
theo... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 588 | 600 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | rw [Iso.inv_comp_eq, ← Category.assoc, Iso.eq_comp_inv, isoZeroCocycles_hom_comp_subtype]
/-- The 0th group cohomology of `A`, defined as the 0th cohomology of the complex of inhomogeneous
cochains, is isomorphic to the invariants of the representation on `A`. -/
def isoH0 : groupCohomology A 0 ≅ H0 A :=
(CochainC... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 815 | 821 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | (μ : Measure (Quotient α_mod_G)) [i : QuotientMeasureEqMeasurePreimage ν μ] {t : Set α}
(fund_dom_t : IsFundamentalDomain G t ν) {U : Set (Quotient α_mod_G)}
(meas_U : MeasurableSet U) : μ U = ν (π ⁻¹' U ∩ t) := by
rw [fund_dom_t.projection_respects_measure (μ := μ), measure_map_restrict_apply ν t meas_U]... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 766 | 780 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.OfNat
/-!
# Cast of natural numbers
This file defines the *can... | | zero => rw [Nat.binCast, Nat.cast_zero]
| succ k =>
rw [Nat.binCast]
by_cases h : (k + 1) % 2 = 0
| Mathlib/Data/Nat/Cast/Defs.lean | 159 | 162 |
/-
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... | induction x using Localization.induction_on
rw [← ofFractionRing_smul, ← ofFractionRing_smul, Localization.smul_mk,
Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]
section IsDomain
| Mathlib/FieldTheory/RatFunc/Basic.lean | 183 | 188 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | #### Convex sets in an ordered space
Relates `Convex` and `OrdConnected`.
-/
| Mathlib/Analysis/Convex/Basic.lean | 547 | 549 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | @[deprecated (since := "2024-11-24")]
alias CompleteLattice.independent_iff_supIndep_univ := iSupIndep_iff_supIndep_univ
alias ⟨iSupIndep.sup_indep_univ, Finset.SupIndep.iSupIndep_of_univ⟩ := iSupIndep_iff_supIndep_univ
| Mathlib/Order/SupIndep.lean | 511 | 514 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... | shiftFunctorAdd' C a₁₂ a₃ a₁₂₃ (by rw [← h₁₂, h₁₂₃]) ≪≫
isoWhiskerRight (shiftFunctorAdd' C a₁ a₂ a₁₂ h₁₂) _ ≪≫ Functor.associator _ _ _ =
shiftFunctorAdd' C a₁ a₂₃ a₁₂₃ (by rw [← h₂₃, ← add_assoc, h₁₂₃]) ≪≫
isoWhiskerLeft _ (shiftFunctorAdd' C a₂ a₃ a₂₃ h₂₃) := by
subst h₁₂ h₂₃ h₁₂₃
ext X
dsi... | Mathlib/CategoryTheory/Shift/Basic.lean | 241 | 248 |
/-
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.... | rw [Functions.apply₂, Term.realize]
refine congr rfl (funext (Fin.cases ?_ ?_))
· simp only [Matrix.cons_val_zero]
· simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
| Mathlib/ModelTheory/Semantics.lean | 109 | 113 |
/-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey
-/
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Nat.Count
/-!
# Periodic Functions on ℕ
This file identifies a few functions on `ℕ` which are periodic, and a... | conv_rhs => rw [← Nat.mod_add_div n a, mul_comm, ← Nat.nsmul_eq_mul, hf.nsmul]
| Mathlib/Data/Nat/Periodic.lean | 33 | 34 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,667 | 1,668 | |
/-
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.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | | Or.inl h => inf_le_of_left_le <| sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le <| sup_le_sup_left (le_inf h (le_refl c)) _
| Mathlib/Order/Lattice.lean | 739 | 740 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... |
section IsFractionRing
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 257 | 258 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | theorem Maximal.eq_of_le (hx : Maximal P x) (hy : P y) (hle : x ≤ y) : x = y :=
hle.antisymm <| hx.2 hy hle
theorem Maximal.eq_of_ge (hx : Maximal P x) (hy : P y) (hge : x ≤ y) : y = x :=
(hx.eq_of_le hy hge).symm
| Mathlib/Order/Minimal.lean | 219 | 223 |
/-
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.Option.Basic
import Batteries.Tactic.Congr
import Mathlib.Data.Set.Basic
import Mathlib.Tactic.Contrapose
/-!
# Partial Equivalences
In this file, ... |
@[simp]
| Mathlib/Data/PEquiv.lean | 221 | 222 |
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