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/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 906 | 908 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | rcases hs with ⟨fx, hx, fy, hy, hxy⟩
rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩
| Mathlib/Data/Set/Image.lean | 1,031 | 1,032 |
/-
Copyright (c) 2022 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Archimedean.Hom
import Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice
/-!
# Conditionally complete linear ordered field... | theorem exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self (ha : 0 < a) (b : β)
(hba : b < inducedMap α β a * inducedMap α β a) : ∃ c ∈ cutMap β (a * a), b < c := by
| Mathlib/Algebra/Order/CompleteField.lean | 244 | 245 |
/-
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.Logic.Function.Iterate
import Mathlib.Order.Monotone.Basic
/-!
# Inequalities on iterates
In this file we prove some inequalities comparing `f^[n] ... | ### Iterates of commuting functions
If `f` and `g` are monotone and commute, then `f x ≤ g x` implies `f^[n] x ≤ g^[n] x`, see
`Function.Commute.iterate_le_of_map_le`. We also prove two strict inequality versions of this lemma,
| Mathlib/Order/Iterate.lean | 142 | 145 |
/-
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.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
/-!
# More basic logic properties
A few more logic le... |
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
| Mathlib/Logic/Lemmas.lean | 38 | 41 |
/-
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.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | lemma realPart.norm_le (x : A) : ‖realPart x‖ ≤ ‖x‖ := by
rw [← inv_mul_cancel_left₀ two_ne_zero ‖x‖, ← AddSubgroup.norm_coe, realPart_apply_coe,
norm_smul, norm_inv, Real.norm_ofNat]
gcongr
exact norm_add_le _ _ |>.trans <| by simp [two_mul]
| Mathlib/Analysis/Complex/Basic.lean | 616 | 621 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... | have ha : a ≠ 0 := ne_of_gt (lt_of_lt_of_le hc hh)
rw [← inv_le_inv₀ (zero_lt_iff.2 ha) hc] at hh
simpa [inv_mul_cancel_left₀ ha, inv_mul_cancel_left₀ hc.ne']
using mul_lt_mul_of_le_of_lt_of_nonneg_of_pos hh h zero_le' (inv_pos.2 hc)
@[deprecated div_le_div_iff_of_pos_right (since := "2024-11-18")]
theorem ... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 168 | 179 |
/-
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.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.Lemmas
/-!
## `norm_num` plugin for scientific notation.
-/
namespace Mathlib
open... | theorem isNat_ofScientific_of_false [DivisionRing α] : {m e nm ne n : ℕ} →
IsNat m nm → IsNat e ne → n = Nat.mul nm ((10 : ℕ) ^ ne) →
IsNat (OfScientific.ofScientific m false e : α) n
| _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => ⟨by
rw [← Rat.cast_ofScientific, ← Rat.ofScientific_eq_ofScientific]
simp only [Na... | Mathlib/Tactic/NormNum/OfScientific.lean | 30 | 37 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | · rw [h, pow_one, moebius_apply_prime hp]
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,019 | 1,019 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... | rw [← abs_map, ← abs_map, ← abs_map]
rfl
theorem lawfulFunctor
(h : ∀ α β : Type u, @Functor.mapConst F _ α _ = Functor.map ∘ Function.const β) :
LawfulFunctor F :=
| Mathlib/Data/QPF/Univariate/Basic.lean | 78 | 83 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Complex.ext_iff.2 <| by simp [ofReal]
| Mathlib/Data/Complex/Basic.lean | 214 | 215 |
/-
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, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... |
theorem isPiSystem_measurableCylinders : IsPiSystem (measurableCylinders α) :=
fun _ hS _ hT _ ↦ inter_mem_measurableCylinders hS hT
| Mathlib/MeasureTheory/Constructions/Cylinders.lean | 324 | 326 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,532 | 1,537 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | theorem eq_rotation_self_iff_angle_eq_zero {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
x = o.rotation θ x ↔ θ = 0 := by rw [← o.rotation_eq_self_iff_angle_eq_zero hx, eq_comm]
/-- A rotation of a vector equals that vector if and only if the vector or the angle is zero. -/
theorem rotation_eq_self_iff (x : V) (θ : Real... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 260 | 265 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.... | lemma isNilpotent_trace_of_isNilpotent {f : M →ₗ[R] M} (hf : IsNilpotent f) :
IsNilpotent (trace R M f) := by
by_cases H : ∃ s : Finset M, Nonempty (Basis s R M)
swap
· rw [LinearMap.trace, dif_neg H]
exact IsNilpotent.zero
obtain ⟨s, ⟨b⟩⟩ := H
classical
rw [trace_eq_matrix_trace R b]
apply Matrix... | Mathlib/LinearAlgebra/Trace.lean | 294 | 307 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals of finsets as finsets
This file provides... | /-- A function `f` from `Finset α` is strictly antitone if and only if `f (insert a s) < f s` for
all `s` and `a ∉ s`. -/
lemma strictAnti_iff_forall_lt_insert : StrictAnti f ↔ ∀ s ⦃a⦄, a ∉ s → f (insert a s) < f s :=
strictMono_iff_forall_lt_insert (β := βᵒᵈ)
| Mathlib/Data/Finset/Interval.lean | 165 | 168 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | the best we can do in general. -/
theorem Adj.card_commonNeighbors_lt_degree {G : SimpleGraph V} [DecidableRel G.Adj] {v w : V}
(h : G.Adj v w) : Fintype.card (G.commonNeighbors v w) < G.degree v := by
classical
rw [← Set.toFinset_card]
apply Finset.card_lt_card
rw [Finset.ssubset_iff]
use w
constructor... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 399 | 407 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | ### Boolean algebras
-/
/-- A Boolean algebra is a bounded distributive lattice with a complement operator `ᶜ` such that
`x ⊓ xᶜ = ⊥` and `x ⊔ xᶜ = ⊤`. For convenience, it must also provide a set difference operation `\`
and a Heyting implication `⇨` satisfying `x \ y = x ⊓ yᶜ` and `x ⇨ y = y ⊔ xᶜ`.
| Mathlib/Order/BooleanAlgebra.lean | 477 | 483 |
/-
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.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.... | theorem isNat_natCast {R} [AddMonoidWithOne R] (n m : ℕ) :
IsNat n m → IsNat (n : R) m := by rintro ⟨⟨⟩⟩; exact ⟨rfl⟩
| Mathlib/Tactic/NormNum/Basic.lean | 104 | 105 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... |
/-- Composition with a fixed zero-preserving homomorphism is itself a zero-preserving homomorphism
on functions. -/
@[simps]
def mapRange.zeroHom (f : ZeroHom M N) : ZeroHom (α →₀ M) (α →₀ N) where
toFun := (mapRange f f.map_zero : (α →₀ M) → α →₀ N)
| Mathlib/Data/Finsupp/Basic.lean | 158 | 163 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have dνκ_finite := (levyProkhorovEDist_lt_top ν κ).ne
convert ENNReal.toReal_mono ?_ <| levyProkhorovEDist_triangle μ ν κ
· simp only [levyProkhorovDist, ENNReal.toReal_add dμν_finite dνκ_finite]
· exact ENNReal.add_ne_top.mpr ⟨dμν_finite, dνκ_finite⟩
/-- A type synonym, to be used for `Measure α`, `FiniteMeas... | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | 184 | 195 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# Rational roo... | haveI inst := Classical.propDecidable
by_cases hr : num A r = 0
· simp_all [nonZeroDivisors.coe_ne_zero]
· refine dvd_of_dvd_mul_left_of_no_prime_factors hr ?_ this
intro q dvd_num dvd_denom_pow hq
apply hq.not_unit
exact num_den_reduced A r dvd_num (hq.dvd_of_dvd_pow dvd_denom_pow)
... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 69 | 86 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
/-!
# Subalgebras in towers of algebras
In this file we prove facts about ... |
@[simp]
| Mathlib/Algebra/Algebra/Subalgebra/Tower.lean | 97 | 98 |
/-
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.RingTheory.DedekindDomain.Ideal
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideals of `R`
inside its f... | Mathlib/RingTheory/ClassGroup.lean | 400 | 409 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals of finsets as finsets
This file provides... | rw [card_Ico_eq_card_Icc_sub_one, card_Icc_finset h]
/-- Cardinality of an `Ioc` of finsets. -/
theorem card_Ioc_finset (h : s ⊆ t) : (Ioc s t).card = 2 ^ (t.card - s.card) - 1 := by
rw [card_Ioc_eq_card_Icc_sub_one, card_Icc_finset h]
| Mathlib/Data/Finset/Interval.lean | 101 | 106 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | exact hsymm x
have I : e '' e.source = e.target := e.image_source_eq_target
have I' : e' '' e'.source = e'.target := e'.image_source_eq_target
| Mathlib/Logic/Equiv/PartialEquiv.lean | 467 | 469 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 761 | 767 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,285 | 1,317 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Data.Set.CoeSort
import Mathlib.T... | -- Isomorphisms in Type and equivalences.
namespace Equiv
universe u
variable {X Y : Type u}
/-- Any equivalence between types in the same universe gives
a categorical isomorphism between those types.
| Mathlib/CategoryTheory/Types.lean | 275 | 283 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | (g : X ⟶ Z) : coprod.inl ≫ coprod.map f g = f ≫ coprod.inl :=
ι_colimMap _ _
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 747 | 750 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... |
theorem prod_univ_units_id_eq_neg_one [CommRing K] [IsDomain K] [Fintype Kˣ] :
∏ x : Kˣ, x = (-1 : Kˣ) := by
classical
have : (∏ x ∈ (@univ Kˣ _).erase (-1), x) = 1 :=
prod_involution (fun x _ => x⁻¹) (by simp)
(fun a => by simp +contextual [Units.inv_eq_self_iff])
(fun a => by simp [@i... | Mathlib/FieldTheory/Finite/Basic.lean | 104 | 111 |
/-
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... | fun _ ha _ _ hab => rpow_lt_rpow_of_exponent_neg ha hab hr
theorem rpow_le_rpow_of_exponent_nonpos {x y : ℝ} (hy : 0 < y) (hxy : y ≤ x) (hz : z ≤ 0) :
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 605 | 607 |
/-
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.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 746 | 749 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... |
theorem inv_comp (r : Rel α β) (s : Rel β γ) : inv (r • s) = inv s • inv r := by
ext x z
| Mathlib/Data/Rel.lean | 131 | 133 |
/-
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... |
lemma map_transitive {r : α → α → Prop} (hr : Transitive r) {f : α → β}
(hf : ∀ x y, f x = f y → r x y) :
| Mathlib/Logic/Relation.lean | 233 | 235 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Jacobson.Ideal
import Mathlib.RingTheory.Ideal.Quo... | rfl
theorem eval_surjective (n : ℕ) : Function.Surjective (eval I M n) := fun x ↦
| Mathlib/RingTheory/AdicCompletion/Basic.lean | 313 | 315 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | apply (measure_mono (μ := ν) B_subset).trans
rw [prob_compl_eq_one_sub isOpen_thickening.measurableSet]
have Tc_mble := (isOpen_thickening (δ := ε.toReal) (E := B)).isClosed_compl.measurableSet
specialize h ε (thickening ε.toReal B)ᶜ ε_gt ε_lt_top Tc_mble
rw [prob_compl_eq_one_sub isOpen_thickening.measurable... | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | 298 | 306 |
/-
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... |
lemma rpow_add_natCast' (hx : 0 ≤ x) (h : y + n ≠ 0) : x ^ (y + n) = x ^ y * x ^ n := by
rw [rpow_add' hx h, rpow_natCast]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 421 | 424 |
/-
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.Polynomial.Mirror
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.Order.Units
import Mathlib.RingTheory.Coprime.Basic
/-!
# Unit... | if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 61 | 64 |
/-
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 | 2,774 | 2,787 | |
/-
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.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | have : N.lcs k ≤ N.incl.range := by
rw [N.range_incl]
apply lcs_le_self
rw [ih, LieSubmodule.comap_bracket_eq _ N.incl _ N.ker_incl this]
theorem lowerCentralSeries_map_eq_lcs : (lowerCentralSeries R L N k).map N.incl = N.lcs k := by
rw [lowerCentralSeries_eq_lcs_comap, LieSubmodule.map_comap_inc... | Mathlib/Algebra/Lie/Nilpotent.lean | 143 | 152 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vectors and ring op... | apply constantCoeff_wittStructureInt p _ _ n
simp only [mul_zero, RingHom.map_mul, constantCoeff_X]
@[simp]
| Mathlib/RingTheory/WittVector/Defs.lean | 265 | 268 |
/-
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` ... | ← map_inf_principal_preimage]; rfl
theorem continuousWithinAt_prod_of_discrete_right [DiscreteTopology β]
{f : α × β → γ} {s : Set (α × β)} {x : α × β} :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨·, x.2⟩) {a | (a, x.2) ∈ s} x.1 := by
rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_pro... | Mathlib/Topology/ContinuousOn.lean | 1,120 | 1,125 |
/-
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... | simp [Complex.ofReal_log hx]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 253 | 254 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.C... | dsimp [bijection]
-- Porting note: added
erw [homEquiv_symm_apply_eq, homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq]
rw [comp_id, comp_id, comp_id, i.map_id, comp_id, unitCompPartialBijective_symm_apply,
unitCompPartialBijective_symm_apply, uncurry_natural_left, uncurry_curry,
| Mathlib/CategoryTheory/Closed/Ideal.lean | 242 | 246 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... | rfl
/-- `Equiv.mulRight₀` as an `OrderIso` on a `LinearOrderedCommGroupWithZero.`. -/
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 191 | 193 |
/-
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.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
/-!
# Remov... | rcases (nhdsWithin_hasBasis nhds_basis_closedBall _).mem_iff.1 hd with ⟨R, hR0, hRs⟩
lift R to ℝ≥0 using hR0.le
replace hc : ContinuousOn f (closedBall c R) := by
refine fun z hz => ContinuousAt.continuousWithinAt ?_
rcases eq_or_ne z c with (rfl | hne)
exacts [hc, (hRs ⟨hz, hne⟩).continuousAt]
exac... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 34 | 43 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | ⟨hf.1, fun i j hij => (hf.2 i j hij).le, fun i => hf.integrable i⟩
theorem submartingale [Preorder E] (hf : Martingale f ℱ μ) : Submartingale f ℱ μ :=
| Mathlib/Probability/Martingale/Basic.lean | 119 | 121 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | if a0 : a = 0 then by simp only [a0, zero_mod]
else by simp only [mod_def, div_self a0, mul_one, sub_self]
@[simp]
theorem mod_one (a : Ordinal) : a % 1 = 0 := by simp only [mod_def, div_one, one_mul, sub_self]
theorem dvd_of_mod_eq_zero {a b : Ordinal} (H : a % b = 0) : b ∣ a :=
⟨a / b, by simpa [H] using (div... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 996 | 1,004 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Probability.Independence.Basic
/-!
# Integration in Probability Theory
Integra... | (hXm : AEMeasurable X μ) (hYm : AEMeasurable Y μ) :
integral μ (X * Y) = integral μ X * integral μ Y := by
have h1 : AEMeasurable (fun a => ENNReal.ofReal (X a)) μ :=
ENNReal.measurable_ofReal.comp_aemeasurable hXm
have h2 : AEMeasurable (fun a => ENNReal.ofReal (Y a)) μ :=
ENNReal.measurable_ofReal... | Mathlib/Probability/Integration.lean | 203 | 218 |
/-
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.Hom.Basic
/-!
# Unbounded lattice homomorphisms
This file defines unbounded lattice homomorphisms. _Bounded_ lattice homomorphisms are defined in
`... | Mathlib/Order/Hom/Lattice.lean | 1,155 | 1,158 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... |
theorem subset (h : t.WellFoundedOn r) (hst : s ⊆ t) : s.WellFoundedOn r :=
h.mono le_rfl hst
| Mathlib/Order/WellFoundedSet.lean | 127 | 130 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,978 | 1,980 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | rcases hy with ⟨ty, rfl⟩
rcases hz with ⟨tz, rfl⟩
rcases lt_trichotomy ty 0 with (hy0 | rfl | hy0)
· rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)
· rw [wbtw_comm (z := x)]
rw [← or_assoc]
exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonpos _ _ hy0.le hz0.le)
· simp
· exact Or.inr (Or.inr ... | Mathlib/Analysis/Convex/Between.lean | 796 | 818 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | castPred i hi = castPred j hj ↔ i = j := by
simp_rw [Fin.ext_iff, le_antisymm_iff, ← le_def, castPred_le_castPred_iff]
theorem castPred_zero' [NeZero n] (h := Fin.ext_iff.not.2 last_pos'.ne) :
| Mathlib/Data/Fin/Basic.lean | 858 | 861 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | rw [geom_sum_succ']
| Mathlib/Algebra/GeomSum.lean | 61 | 61 |
/-
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.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,533 | 1,537 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | exact Equiv.Perm.IsCycleOn.exists_pow_eq (b := y) (f.isCycleOn_support_cycleOf_aux x)
(by rw [mem_support_cycleOf_iff'_aux hx]) (by rwa [mem_support_cycleOf_iff'_aux hx])
| Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 231 | 232 |
/-
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 | 434 | 436 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | · rw [descFactorial_eq_zero_iff_lt.mpr (show n < k by omega), Nat.mul_zero]
/-- Avoid in favor of `Nat.factorial_mul_descFactorial` if you can. ℕ-division isn't worth it. -/
theorem descFactorial_eq_div {n k : ℕ} (h : k ≤ n) : n.descFactorial k = n ! / (n - k)! := by
apply Nat.mul_left_cancel (n - k).factorial_p... | Mathlib/Data/Nat/Factorial/Basic.lean | 387 | 393 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jung Tao Cheng, Christian Merten, Andrew Yang
-/
import Mathlib.LinearAlgebra.TensorProduct.RightExactness
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Gene... | (ofBijectiveAlgebraMap h).dimension = 0 := by
simp_rw [dimension, ofBijectiveAlgebraMap, Generators.ofSurjectiveAlgebraMap,
Generators.ofSurjective, Nat.card_eq_fintype_card, Fintype.card_ofIsEmpty]
variable (R) in
/-- The canonical `R`-presentation of `R` with no generators and no relations. -/
noncomputabl... | Mathlib/RingTheory/Presentation.lean | 173 | 221 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | ext x
refine ⟨fun h => ?_, fun hx t h => ?_⟩ <;> contrapose! h
· simp only [mem_sInter, mem_setOf_eq, not_forall, exists_prop]
exact ⟨{x}ᶜ, by simpa using h, by simp⟩
· exact hx.mono fun i hi => ⟨hi.1, fun hit => h (hit hi.2)⟩
theorem cofinite.bliminf_set_eq : bliminf s cofinite p = { x | { n | p n ∧ x ∉ s... | Mathlib/Order/LiminfLimsup.lean | 705 | 712 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | theorem hasseDeriv_one' : hasseDeriv 1 f = derivative f := by
simp only [hasseDeriv_apply, derivative_apply, ← C_mul_X_pow_eq_monomial, Nat.choose_one_right,
(Nat.cast_commute _ _).eq]
@[simp]
| Mathlib/Algebra/Polynomial/HasseDeriv.lean | 93 | 97 |
/-
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.RCLike.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
/-!
# Measurability of the basic `RCLike` functions
-/
nonco... | ((RCLike.measurable_ofReal.comp_aemeasurable him).mul_const RCLike.I)
exact (RCLike.re_add_im _).symm
end
| Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean | 73 | 77 |
/-
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 | 1,127 | 1,129 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | constructor
· -- We want to use that fact that `0 ≤ s_mod p (p-2) < 2^p - 1`
| Mathlib/NumberTheory/LucasLehmer.lean | 169 | 170 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | NodupKeys l :=
(nodupKeys_cons.1 h).2
| Mathlib/Data/List/Sigma.lean | 102 | 103 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 390 | 394 | |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheo... | -- NB: this `simpa` is quite slow if un-squeezed
simpa only [LSeriesHasSum, term_convolution'] using (hf.mul hg hsum).tsum_fiberwise _
/-- The L-series of the convolution product `f ⍟ g` of two sequences `f` and `g`
equals the product of their L-series, assuming both L-series converge. -/
lemma LSeries_convolution... | Mathlib/NumberTheory/LSeries/Convolution.lean | 139 | 146 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.Order.Floor.Defs
/-!
# Computable Continued Fractions
## Summary
We formalise the standa... | -/
protected def seq1 (v : K) : Stream'.Seq1 <| IntFractPair K :=
⟨IntFractPair.of v, -- the head
| Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean | 159 | 161 |
/-
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.Defs
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodu... |
@[deprecated (since := "2025-04-28")]
alias End_algebraMap_isUnit_inv_apply_eq_iff' := End.algebraMap_isUnit_inv_apply_eq_iff'
end
end Module
namespace LinearMap
| Mathlib/Algebra/Algebra/Basic.lean | 212 | 220 |
/-
Copyright (c) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
import Mathlib.GroupTheory.PGroup
import Mathlib.LinearAlgebra.Quotient.Defs
/-!
# Torsion groups
This file defines torsion groups, i.e. groups where all elements hav... | Mathlib/GroupTheory/Torsion.lean | 453 | 457 | |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.CompHausLike.Limits
import Mathlib.Topology.Category.LightProfinite.Basic
/-!
# Explicit limits and colimits
This file applies ... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 60 | 63 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,416 | 1,421 | |
/-
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.ComplexShape
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.CategoryTheory.GradedObject.Trifunctor
/-! Signs in constructions on homologi... | TensorSigns.add_rel _ _ _ hpq
@[simp]
| Mathlib/Algebra/Homology/ComplexShapeSigns.lean | 131 | 133 |
/-
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... | theorem update_comp_eq_of_not_mem_range {α : Sort*} {β : Type*} {γ : Sort*} [DecidableEq β]
(g : β → γ) {f : α → β} {i : β} (a : γ) (h : i ∉ Set.range f) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_not_mem_range' g a h
theorem insert_injOn (s : Set α) : sᶜ.InjOn fun a => insert a s := fun _a ha _ _ =>
| Mathlib/Data/Set/Function.lean | 1,150 | 1,154 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.Grp.ZModuleEquivalence
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra... | obtain rfl | h0 := eq_or_ne m 0
· refine ⟨0, fun n hn ↦ ?_⟩
rw [Submodule.span_zero_singleton] at hn
subst hn
exact (map_zero g).symm
let gₘ := g ⟨m, Submodule.subset_span (Set.mem_singleton _)⟩
refine ⟨LinearMap.toSpanSingleton ℤ A (DivisibleBy.div gₘ m), fun n hn ↦ ?_⟩
rcases Submodule.mem_span_... | Mathlib/Algebra/Category/Grp/Injective.lean | 47 | 58 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... | theorem condition (a) : I.fst a ≫ K.π (J.fst a) = I.snd a ≫ K.π (J.snd a) := by
rw [← K.snd_app_right, ← K.fst_app_right]
| Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 581 | 582 |
/-
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.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | Mathlib/Algebra/Lie/Weights/Basic.lean | 792 | 796 | |
/-
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... | · rw [hf, eraseLead_zero, support_zero, card_empty]
· rw [← card_support_eraseLead_add_one hf, add_tsub_cancel_right]
| Mathlib/Algebra/Polynomial/EraseLead.lean | 110 | 112 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.LiftingProperties.Basic
import Mathlib.CategoryTheory.Adjunction.Basic
/-!
# Lifting properties and adjunction
In this file, we obtain `Adjunct... | simp only [Adjunction.homEquiv_unit, assoc, ← F.map_comp, sq.w]
rw [F.map_comp, Adjunction.unit_naturality_assoc]⟩
variable (sq : CommSq u (G.map i) p v) (adj : G ⊣ F)
| Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean | 40 | 43 |
/-
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.Data.Finset.Lattice.Prod
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Setoid.Basic
import Mathlib.Order.Atoms
impor... | fun ps ↦ if h : ps.sup id = s ∧ ⊥ ∉ ps ∧ ps.SupIndep id then ⟨ps, h.2.2, h.1, h.2.1⟩ else ⊤
complete P := by
refine mem_image.mpr ⟨P.parts, ?_, ?_⟩
· rw [mem_powerset]; intro p hp; rw [mem_powerset]; exact P.le hp
· simp [P.supIndep, P.sup_parts, P.not_bot_mem, -bot_eq_empty]
| Mathlib/Order/Partition/Finpartition.lean | 269 | 274 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... | rcases ih (fun _ hg => hw _ (List.mem_cons_of_mem _ hg)) with
⟨w', hw'prod, hw'map⟩
refine ⟨cons g w' ?_ ?_, ?_⟩
· rwa [Word.fstIdx, ← List.head?_map, hw'map, List.head?_map]
· exact hw _ List.mem_cons_self
· simp [hw'prod, hw'map]
/-- For any word `w` in the coproduct,
if `w` is reduced (i.e... | Mathlib/GroupTheory/PushoutI.lean | 626 | 638 |
/-
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | IsComplete (univ : Set α) := fun f hf _ => by
rcases CompleteSpace.complete hf with ⟨x, hx⟩
exact ⟨x, mem_univ x, hx⟩
instance CompleteSpace.prod [UniformSpace β] [CompleteSpace α] [CompleteSpace β] :
CompleteSpace (α × β) where
| Mathlib/Topology/UniformSpace/Cauchy.lean | 366 | 371 |
/-
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, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 1,600 | 1,605 | |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=
fun h x₁ x₂ y' ↦ by
rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]
apply h
| Mathlib/SetTheory/Surreal/Multiplication.lean | 202 | 205 |
/-
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.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... |
section cast
theorem natCast_coeff_zero {n : ℕ} {R : Type*} [Semiring R] : (n : R[X]).coeff 0 = n := by
simp only [coeff_natCast_ite, ite_true]
@[norm_cast]
theorem natCast_inj {m n : ℕ} {R : Type*} [Semiring R] [CharZero R] :
(↑m : R[X]) = ↑n ↔ m = n := by
constructor
| Mathlib/Algebra/Polynomial/Coeff.lean | 334 | 343 |
/-
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... |
theorem rootMultiplicity_eq_zero {p : R[X]} {x : R} (h : ¬IsRoot p x) : rootMultiplicity x p = 0 :=
rootMultiplicity_eq_zero_iff.2 fun h' => (h h').elim
@[simp]
| Mathlib/Algebra/Polynomial/Div.lean | 629 | 633 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | | inl h => rw [h.neg_one_pow, if_pos h]
| Mathlib/Algebra/Ring/Parity.lean | 348 | 348 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | (h : smul n x = y) : n • x = y := h
theorem unfold_smulg {α} [AddCommGroup α] (n : ℕ) (x y : α)
| Mathlib/Tactic/Abel.lean | 291 | 293 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Module.LinearMap.Defs
import Mathlib.RingTheory.HahnSeries.Basic
import Mathlib.Algebra.BigOperators.Group.... | induction n with
| zero => simp
| succ n ih => simp [add_nsmul, ih]
@[deprecated (since := "2025-01-31")] alias nsmul_coeff := coeff_nsmul
| Mathlib/RingTheory/HahnSeries/Addition.lean | 74 | 79 |
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# p-groups
This file contains a proof that if `G` is a `p`-group ac... | IsPGroup p (H.comap K.subtype) :=
hH.comap_of_injective K.subtype Subtype.coe_injective
theorem to_sup_of_normal_right {H K : Subgroup G} (hH : IsPGroup p H) (hK : IsPGroup p K)
[K.Normal] : IsPGroup p (H ⊔ K : Subgroup G) := by
| Mathlib/GroupTheory/PGroup.lean | 264 | 268 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.RingTheory.Norm.Defs
/-!
# N... | theorem associated_norm_prod_smith [Fintype ι] (b : Basis ι R S) {f : S} (hf : f ≠ 0) :
Associated (Algebra.norm R f) (∏ i, smithCoeffs b _ (span_singleton_eq_bot.not.2 hf) i) := by
have hI := span_singleton_eq_bot.not.2 hf
let b' := ringBasis b (span {f}) hI
classical
rw [← Matrix.det_diagonal, ← LinearMap... | Mathlib/LinearAlgebra/FreeModule/Norm.lean | 30 | 50 |
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | rwa [Monic, Polynomial.leadingCoeff, natDegree, (lt_or_eq_of_le H1).resolve_left H]
theorem monic_X_pow_add {n : ℕ} (H : degree p < n) : Monic (X ^ n + p) :=
monic_of_degree_le n
(le_trans (degree_add_le _ _) (max_le (degree_X_pow_le _) (le_of_lt H)))
| Mathlib/Algebra/Polynomial/Monic.lean | 84 | 88 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
@[simp] theorem one_le_encard_iff_nonempty : 1 ≤ s.encard ↔ s.Nonempty := by
rw [nonempty_iff_ne_empty, Ne, ← encard_eq_zero, ENat.one_le_iff_ne_zero]
| Mathlib/Data/Set/Card.lean | 174 | 177 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... | mongePointWeightsWithCircumcenter] <;>
rw [add_tsub_assoc_of_le (by decide : 1 ≤ 2), (by decide : 2 - 1 = 1)]
· rw [if_pos (mem_univ _), sub_zero, add_zero, card_fin]
-- Porting note: replaced
-- have hn3 : (n + 2 + 1 : ℝ) ≠ 0 := mod_cast Nat.succ_ne_zero _
have hn3 : (n + 2 + 1 : ℝ) ≠ 0 := by... | Mathlib/Geometry/Euclidean/MongePoint.lean | 130 | 154 |
/-
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.Induced
import Mathlib.MeasureTheory.OuterMeasure.AE
import Mathlib.Order.Filter.CountableInter
/-!
# Measu... | (hfin : ∀ i ∈ s, μ (f i) < ∞) : μ (⋃ i ∈ s, f i) < ∞ := by
convert (measure_biUnion_finset_le (μ := μ) hs.toFinset f).trans_lt _ using 3
· ext
rw [Finite.mem_toFinset]
| Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 221 | 224 |
/-
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... | Mathlib/Topology/Irreducible.lean | 331 | 338 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | obtain ⟨t, ht⟩ := hk s hsup
-- certainly every element of t is below something in s, since ↑t ⊆ s.
have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s
exact ⟨x, ⟨hxs, le_trans ht.r... | Mathlib/Order/CompactlyGenerated/Basic.lean | 110 | 149 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... | theorem rank_le_of_surjective_injective (i : R → R') (j : M →+ M₁)
(hi : Surjective i) (hj : Injective j)
(hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) :
Module.rank R M ≤ Module.rank R' M₁ := by
simpa only [lift_id] using lift_rank_le_of_surjective_injective i j hi hj hc
| Mathlib/LinearAlgebra/Dimension/Basic.lean | 163 | 167 |
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