Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingT... | intro x hx
rw [mem_nonZeroDivisors_iff_ne_zero]
intro hx'
apply @zero_ne_one S
rw [← (algebraMap R S).map_one, ← @mk'_one R _ M, @comm _ Eq, mk'_eq_zero_iff]
exact ⟨⟨x, hx⟩, by simp [hx']⟩
instance {R : Type*} [CommRing R] [IsDomain R] (p : Ideal R) [p.IsPrime] :
IsFractionRing (Localization.AtPrime p)... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 281 | 292 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... |
theorem Balanced.sub (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t) : Balanced 𝕜 (s - t) := by
simp_rw [sub_eq_add_neg]
| Mathlib/Analysis/LocallyConvex/Basic.lean | 138 | 140 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... |
theorem isLeast_Ici : IsLeast (Ici a) a :=
⟨left_mem_Ici, fun _ => id⟩
| Mathlib/Order/Bounds/Basic.lean | 392 | 394 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Ana... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 387 | 435 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Bounds.Image
import Mathlib.Order.Hom.Set
/-!
# Order isomorphisms and bounds.
-/
open Set
namespace OrderIso
variable {α β ... |
theorem isGLB_preimage' {s : Set β} {x : β} : IsGLB (f ⁻¹' s) (f.symm x) ↔ IsGLB s x :=
| Mathlib/Order/Bounds/OrderIso.lean | 53 | 54 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Nat
import Mathli... | rw [← Ico_bot, image_cast_int_Ico]; rfl
| Mathlib/Data/Nat/Cast/SetInterval.lean | 43 | 44 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 540 | 541 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring.Finset
import ... | lemma sum_range_add_choose (n k : ℕ) :
∑ i ∈ Finset.range (n + 1), (i + k).choose k = (n + k + 1).choose (k + 1) := by
rw [← sum_Icc_choose, range_eq_Ico]
convert (sum_map _ (addRightEmbedding k) (·.choose k)).symm using 2
rw [map_add_right_Ico, zero_add, add_right_comm, Nat.Ico_succ_right]
end Nat
| Mathlib/Data/Nat/Choose/Sum.lean | 143 | 150 |
/-
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.Data.Set.Finite.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.SuccPred.Ba... | · show (Iic i).inf (p ∘ f) ↔ _
simp [Finset.inf_eq_iInf]
· intro x y
rw [hp]
| Mathlib/Order/PartialSups.lean | 63 | 66 |
/-
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 | 1,136 | 1,138 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generali... | Mathlib/Topology/Gluing.lean | 534 | 545 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | /-- This is a special case of `WithTop.sub_top` in the `ENNReal` namespace -/
theorem sub_top : a - ∞ = 0 := WithTop.sub_top
@[simp] theorem sub_eq_top_iff : a - b = ∞ ↔ a = ∞ ∧ b ≠ ∞ := WithTop.sub_eq_top_iff
lemma sub_ne_top_iff : a - b ≠ ∞ ↔ a ≠ ∞ ∨ b = ∞ := WithTop.sub_ne_top_iff
-- This is unsafe because we coul... | Mathlib/Data/ENNReal/Operations.lean | 300 | 306 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,160 | 2,165 | |
/-
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.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
/-!
# Angles between vectors
This fil... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 371 | 374 | |
/-
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.Topology.MetricSpace.Lipschitz
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
/-!
# Hölder continuous functions
In this file we define Höl... |
theorem dist_le (hf : HolderOnWith C r f s) (hx : x ∈ s) (hy : y ∈ s) :
dist (f x) (f y) ≤ C * dist x y ^ (r : ℝ) :=
hf.dist_le_of_le hx hy le_rfl
end HolderOnWith
| Mathlib/Topology/MetricSpace/Holder.lean | 237 | 242 |
/-
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.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | @[simp] lemma card_eq_top : card α = ⊤ ↔ Infinite α := by simp [card, aleph0_le_mk_iff]
@[simp] theorem card_lt_top_of_finite [Finite α] : card α < ⊤ := by simp [card]
| Mathlib/SetTheory/Cardinal/Finite.lean | 279 | 281 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Fintype.Order
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Tactic.Linarith
/-!
# Acyclic grap... | choose f hf hf' using (hG.existsUnique_path · default)
refine Eq.symm <| Finset.card_bij
(fun w hw => ((f w).firstDart <| ?notNil).edge)
(fun a ha => ?memEdges) ?inj ?surj
case notNil => exact not_nil_of_ne (by simpa using hw)
case memEdges => simp
case inj =>
intros a ha b hb h
wl... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 160 | 207 |
/-
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... | Mathlib/Data/Set/Lattice.lean | 1,922 | 1,923 | |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.Data.PNat.Prime
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib... | replace hζ : IsPrimitiveRoot ζ (2 ^ k : ℕ+) := by simp [hζ]
obtain ⟨k₁, hk₁⟩ := exists_eq_succ_of_ne_zero (lt_of_lt_of_le zero_lt_two hk).ne.symm
simpa [hk₁, show ((2 : ℕ+) : ℕ) = 2 from rfl] using sub_one_norm_eq_eval_cyclotomic hζ this hirr
/-- If `Irreducible (cyclotomic (p ^ (k + 1)) K)` (in particular for `... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 504 | 519 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | (i : IsColimit c) : BinaryBicone X c.pt where
pt := Y
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 475 | 476 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 555 | 563 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Probability measures
Th... | (mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)) : Tendsto μs F (𝓝 μ) := by
rw [tendsto_iff_forall_testAgainstNN_tendsto]
exact fun f ↦
tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass μs_lim mass_lim f
/-- If finite measures themselves converge weakly to a nonzero limit me... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 498 | 503 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.AEMeasurable
/-!
# Typeclasses for measurability of operations
In this file we define classes `MeasurableMul` etc and prove d... | @[to_additive]
instance Pi.measurableInv {ι : Type*} {α : ι → Type*} [∀ i, Inv (α i)]
[∀ i, MeasurableSpace (α i)] [∀ i, MeasurableInv (α i)] : MeasurableInv (∀ i, α i) :=
⟨measurable_pi_iff.mpr fun i => (measurable_pi_apply i).inv⟩
@[to_additive]
theorem MeasurableSet.inv {s : Set G} (hs : MeasurableSet s) : Me... | Mathlib/MeasureTheory/Group/Arithmetic.lean | 407 | 416 |
/-
Copyright (c) 2024 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.ContinuousMap.CocompactMap
import Mathlib.Topology.MetricSpace.Bounded
/-!
# Cocompact maps in normed g... | theorem CocompactMapClass.norm_le [ProperSpace F] [FunLike 𝓕 E F] [CocompactMapClass 𝓕 E F]
(ε : ℝ) : ∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖ := by
have h := cocompact_tendsto f
rw [tendsto_def] at h
specialize h (Metric.closedBall 0 ε)ᶜ (mem_cocompact_of_closedBall_compl_subset 0 ⟨ε, rfl.subset⟩)
rcases cl... | Mathlib/Analysis/Normed/Group/CocompactMap.lean | 30 | 40 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | I.add_pow_add_pred_mem_of_pow_mem_of_commute ha hb (Commute.all ..)
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
s.sum ^ (Multiset.card s * n + 1) ∈
span ((s.map fun (x : α) ↦ x ^ (n + 1)).toFinset : Set α) := by
| Mathlib/RingTheory/Ideal/Basic.lean | 107 | 111 |
/-
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 | 1,772 | 1,779 | |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | simp_rw [div_eq_mul_inv, ← ENNReal.tsum_mul_right, toOuterMeasure_apply]
refine tsum_congr fun x => ?_
| Mathlib/Probability/Distributions/Uniform.lean | 371 | 372 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... |
section IsUniformGroup
variable [UniformSpace G] [IsUniformGroup G]
@[to_additive]
theorem cauchySeq_finset_iff_nat_tprod_vanishing {f : ℕ → G} :
(CauchySeq fun s : Finset ℕ ↦ ∏ n ∈ s, f n) ↔
∀ e ∈ 𝓝 (1 : G), ∃ N : ℕ, ∀ t ⊆ {n | N ≤ n}, (∏' n : t, f n) ∈ e := by
refine cauchySeq_finset_iff_tprod_vanishi... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 273 | 285 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finite sets in `Option α`
In this file we define
* `Option.t... | theorem some_mem_insertNone {s : Finset α} {a : α} : some a ∈ insertNone s ↔ a ∈ s := by simp
lemma none_mem_insertNone {s : Finset α} : none ∈ insertNone s := by simp
| Mathlib/Data/Finset/Option.lean | 70 | 72 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | variable [DistribLattice α] [LocallyFiniteOrder α] {a b c : α}
| Mathlib/Order/Interval/Finset/Basic.lean | 1,015 | 1,016 |
/-
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.Group.Support
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered an... | {g : Γ} (h : x.coeff g ≠ 0) : x.orderTop ≤ g := by
rw [orderTop_of_ne (ne_zero_of_coeff_ne_zero h), WithTop.coe_le_coe]
exact Set.IsWF.min_le _ _ ((mem_support _ _).2 h)
@[simp]
theorem orderTop_single (h : r ≠ 0) : (single a r).orderTop = a :=
(orderTop_of_ne (single_ne_zero h)).trans
(WithTop.coe_inj.m... | Mathlib/RingTheory/HahnSeries/Basic.lean | 283 | 290 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.Probability.Martingale.Upcrossing
/-!
# Mar... | theorem Submartingale.exists_ae_tendsto_of_bdd [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) : ∀ᵐ ω ∂μ, ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) := by
filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_atTop_of_eLpNorm_bdd one_ne_zero
(fun n => (hf.stronglyMeas... | Mathlib/Probability/Martingale/Convergence.lean | 186 | 190 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | S₁.sc' h₁ i j k ≅ S₂.sc' h₂ i j k where
hom := sc'Map e.hom h₁ h₂ i j k
inv := sc'Map e.inv h₂ h₁ i j k
hom_inv_id := by ext <;> dsimp <;> simp
| Mathlib/Algebra/Homology/ExactSequence.lean | 159 | 162 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | Mathlib/Topology/Connected/Basic.lean | 1,013 | 1,023 | |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | /-- Continuations as in `ToPartrec.Cont` but with the data removed. This is done because we want
| Mathlib/Computability/TMToPartrec.lean | 174 | 174 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.DedekindD... | /-- `eval` is an additive homomorphism except when a denominator evaluates to `0`.
| Mathlib/FieldTheory/RatFunc/AsPolynomial.lean | 135 | 135 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Yury Kudryashov
-/
import Mathlib.Data.Option.Basic
import Mathlib.Topology.Separation.Regular
/-!
# Paracompact topological spaces
A topological space `X` is said to b... | -- For technical reasons we prepend two empty sets to the sequence `CompactExhaustion.choice X`
set K' : CompactExhaustion X := CompactExhaustion.choice X
set K : CompactExhaustion X := K'.shiftr.shiftr
set Kdiff := fun n ↦ K (n + 1) \ interior (K n)
-- Now we restate some properties of `CompactExha... | Mathlib/Topology/Compactness/Paracompact.lean | 196 | 244 |
/-
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... | p := π c₂ c₃ c₂₃
q := π c₁ c₂₃ c
hpq := fun ⟨i₁, i₂, i₃⟩ ↦ (assoc c₁ c₂ c₃ c₁₂ c₂₃ c i₁ i₂ i₃).symm
| Mathlib/Algebra/Homology/ComplexShapeSigns.lean | 254 | 256 |
/-
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.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... |
/-- The tsupport of a partition of unity at a point `x₀` as a `Finset`.
This is the set of `i : ι` such that `x₀ ∈ tsupport f i`. -/
def fintsupport (x₀ : X) : Finset ι :=
(ρ.finite_tsupport x₀).toFinset
| Mathlib/Topology/PartitionOfUnity.lean | 229 | 234 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Order.Field
impo... | rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.2.1
theorem nonuniformWitnesses_spec (h : ¬G.IsUniform ε s t) :
| Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 142 | 145 |
/-
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 | 721 | 722 | |
/-
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.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... |
namespace Finite
| Mathlib/Data/Fintype/Card.lean | 296 | 298 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | map_zero' := span_empty
map_one' := one_eq_span.symm
map_add' := span_union
map_mul' s t := by simp_rw [SetSemiring.down_mul, span_mul_span]
| Mathlib/Algebra/Algebra/Operations.lean | 689 | 693 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... |
variable {X : Type*} [TopologicalSpace X] {a : X} {s : Set X} {f : X → G₀}
@[fun_prop]
nonrec theorem ContinuousAt.zpow₀ (hf : ContinuousAt f a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m) :
ContinuousAt (fun x => f x ^ m) a :=
hf.zpow₀ m h
| Mathlib/Topology/Algebra/GroupWithZero.lean | 328 | 334 |
/-
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.Enum
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
/-!
### Topology of ordinals
We prov... | apply ((mem_closure_tfae a s).out 0 5).trans
simp_rw [exists_prop]
theorem mem_iff_iSup_of_isClosed (hs : IsClosed s) :
| Mathlib/SetTheory/Ordinal/Topology.lean | 132 | 135 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Convex.Deriv
/-!
# The function `x ↦ ... | by_cases hx : x = 0
· rw [hx, inv_zero]
exact deriv_zero_of_not_differentiableAt
(fun h ↦ not_continuousAt_deriv_mul_log_zero h.continuousAt)
| Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | 113 | 116 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | (div_sub_div_same _ _ _).symm
/-- See `inv_sub_inv` for the more convenient version when `K` is commutative. -/
theorem inv_sub_inv' {a b : K} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ - b⁻¹ = a⁻¹ * (b - a) * b⁻¹ :=
let _ := invertibleOfNonzero ha; let _ := invertibleOfNonzero hb; invOf_sub_invOf a b
| Mathlib/Algebra/Field/Basic.lean | 109 | 114 |
/-
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.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... |
open MeasureTheory
| Mathlib/Analysis/BoxIntegral/Basic.lean | 328 | 329 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... | · exact nth_mem a h'
· simp only [not_forall, not_lt] at h'
have h'b : ∃ hf : (setOf p).Finite, #hf.toFinset ≤ b := by
rcases h' with ⟨hf, ha⟩
exact ⟨hf, ha.trans hab⟩
have ha0 : nth p a = 0 := by simp [nth_eq_zero, h']
have hb0 : nth p b = 0 := by simp [nth_eq_zero, h'b]
| Mathlib/Data/Nat/Nth.lean | 269 | 275 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | | .positive pa, .positive pb => pure (.positive q(div_pos $pa $pb))
| .positive pa, .nonnegative pb => pure (.nonnegative q(div_nonneg_of_pos_of_nonneg $pa $pb))
| Mathlib/Algebra/Order/Field/Basic.lean | 712 | 713 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 444 | 446 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
/-!
# Additional lemmas about sum types
Most of the former contents ... | variable {r : α → γ → Prop} {s : β → δ → Prop} {x : α ⊕ β} {y : γ ⊕ δ}
{a : α} {b : β} {c : γ} {d : δ}
| Mathlib/Data/Sum/Basic.lean | 166 | 168 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
/-!
# The rational numbers form a linear ordered field
This f... | Mathlib/Algebra/Order/Ring/Rat.lean | 50 | 59 | |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 897 | 898 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | (Set.image_preimage_eq_inter_range (f := f.base) (t := U.1)).symm ▸
Set.subset_inter_iff.mpr ⟨fun _ h => h, hU⟩).op :=
PresheafedSpace.IsOpenImmersion.app_invApp f U
instance ofRestrict {X : TopCat} (Y : SheafedSpace C) {f : X ⟶ Y.carrier}
(hf : IsOpenEmbedding f) : IsOpenImmersion ... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 809 | 822 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | gauge s x < ε := by
have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
have h_gauge_lt := gauge_lt_one_of_mem_of_isOpen hs₂ this
| Mathlib/Analysis/Convex/Gauge.lean | 366 | 368 |
/-
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 | 814 | 814 | |
/-
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... | P h := by
induction h with
| single h => exact base h
| tail hab hbc h_ih => exact ih hab (single hbc) h_ih (base hbc)
theorem trans_right (hab : ReflTransGen r a b) (hbc : TransGen r b c) : TransGen r a c := by
induction hbc with
| single hbc => exact tail' hab hbc
| tail _ hcd hac => exact hac.tail h... | Mathlib/Logic/Relation.lean | 415 | 423 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 932 | 933 | |
/-
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Module.Submodule.... | /-! ### Linear equivalences -/
| Mathlib/Algebra/Module/Submodule/Ker.lean | 275 | 276 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | simp only [isNClique_iff, card_eq_one, and_iff_right_iff_imp]; rintro ⟨a, rfl⟩; simp
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 240 | 241 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
/-!
# Multivariate quotients of polynomial functors.
Basic definition of multivariate QPF. QPFs form a co... | rw [liftP_iff]
refine ⟨a, f, haf.symm, ?_⟩
intro i u
exact mem_image_of_mem _ (mem_univ _)
exact h this
intro h p; rw [liftP_iff]
rintro ⟨a, f, xeq, h'⟩
rcases h a f xeq.symm with ⟨i, _, hi⟩
rw [← hi]; apply h'
theorem supp_eq {α : TypeVec n} {i} (x : F α) :
supp x i = { u | ∀ a... | Mathlib/Data/QPF/Multivariate/Basic.lean | 160 | 173 |
/-
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 oangle_rotation_self_left {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
o.oangle (o.rotation θ x) x = -θ := by simp [hx]
/-- A vector has an angle of `θ` from the rotation of that vector by `θ`. -/
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 217 | 221 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... |
/-- When `ι` is finite, multilinear maps on `R^ι` with values in `M₂` are in bijection with `M₂`,
as such a multilinear map is completely determined by its value on the constant vector made of ones.
We register this bijection as a linear equivalence in `MultilinearMap.piRingEquiv`. -/
protected def piRingEquiv [Fintyp... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,351 | 1,370 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 550 | 551 | |
/-
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... | simp only [LinearMap.prodMap_apply, LinearMap.id_coe, id, LinearMap.neg_apply,
Prod.mk_inj] at h
rw [mem_graph_iff] at hx' ⊢
rcases hx' with ⟨y, hy, hx'⟩
use y
rw [← h.1, ← h.2]
simp [hy, hx']
| Mathlib/LinearAlgebra/LinearPMap.lean | 744 | 750 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | ‖EuclideanSpace.single i (a : 𝕜)‖₊ = ‖a‖₊ :=
PiLp.nnnorm_equiv_symm_single 2 (fun _ => 𝕜) i a
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 282 | 283 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | · rw [Measure.restrict_prod_eq_prod_univ]
exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm
· rw [Measure.restrict_prod_eq_prod_univ]
exact
(Measure.restrict_eq_self _
(regionBetween_subset (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)).symm
/-- The region between two ... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 533 | 557 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | map_smul' c f := funext <| by simp
invFun r :=
⟨interpolate s v fun x => if hx : x ∈ s then r ⟨x, hx⟩ else 0,
mem_degreeLT.2 <| degree_interpolate_lt _ hvs⟩
| Mathlib/LinearAlgebra/Lagrange.lean | 376 | 379 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... |
variable [NormedField 𝕜]
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 69 | 72 |
/-
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.BigOperators.Finsupp.Fin
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.MvPolynomial.Rename... | coeff n f := by
induction' f using MvPolynomial.induction_on' with j r p q hp hq generalizing n
| Mathlib/Algebra/MvPolynomial/Equiv.lean | 377 | 378 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Module.NatInt
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.FreeGroup.Basic
/-!
# Free abelian groups
The free abelian group on ... | protected theorem unique (g : FreeAbelianGroup α →+ β) (hg : ∀ x, g (of x) = f x) {x} :
g x = lift f x :=
DFunLike.congr_fun (lift.symm_apply_eq.mp (funext hg : g ∘ of = f)) _
/-- See note [partially-applied ext lemmas]. -/
@[ext high]
protected theorem ext (g h : FreeAbelianGroup α →+ β) (H : ∀ x, g (of x) = h ... | Mathlib/GroupTheory/FreeAbelianGroup.lean | 129 | 135 |
/-
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... | alias ⟨IsBlock.smul_eq_smul_or_disjoint, _⟩ := isBlock_iff_smul_eq_smul_or_disjoint
/-- A fixed block is a block. -/
@[to_additive "A fixed block is a block."]
| Mathlib/GroupTheory/GroupAction/Blocks.lean | 211 | 214 |
/-
Copyright (c) 2020 Google LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Wong
-/
import Mathlib.Data.List.Induction
/-!
# Palindromes
This module defines *palindromes*, lists which are equal to their reverse.
The main result is the `Palindrome` inducti... |
theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by
refine bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) ?_
| Mathlib/Data/List/Palindrome.lean | 50 | 52 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Yury Kudryashov
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Nat.ModEq
/-!
# Pigeonhole principles
Given pigeons ... | if the total weight of a finite set of pigeons is less than `n • b`, and they are sorted into `n`
pigeonholes, then for some pigeonhole, the total weight of the pigeons in this pigeonhole is less
than `b`. -/
| Mathlib/Combinatorics/Pigeonhole.lean | 115 | 117 |
/-
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.Real
/-!
# Power function... |
theorem rpow_inv_lt_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ z⁻¹ < y ↔ x < y ^ z := by
simp only [← not_le, le_rpow_inv_iff hz]
theorem lt_rpow_inv_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x < y ^ z⁻¹ ↔ x ^ z < y := by
nth_rw 1 [← rpow_one x]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 749 | 754 |
/-
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... | · linarith [pi_pos, arg_le_pi x]
lemma cpow_inv_two_im_eq_neg_sqrt {x : ℂ} (hx : x.im < 0) :
(x ^ (2⁻¹ : ℂ)).im = -sqrt ((‖x‖ - x.re) / 2) := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 1,009 | 1,012 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Kalle Kytölä
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakBilin
/-!
# Polar set
I... | /-- The polar set is closed in the weak topology induced by `B.flip`. -/
theorem polar_weak_closed (s : Set E) : IsClosed[WeakBilin.instTopologicalSpace B.flip]
(B.polar s) := by
rw [polar_eq_iInter]
refine isClosed_iInter fun x => isClosed_iInter fun _ => ?_
| Mathlib/Analysis/LocallyConvex/Polar.lean | 123 | 127 |
/-
Copyright (c) 2024 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yaël Dillies
-/
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Sums in `WithTop`
This file proves results about ... | lemma sum_ne_top : ∑ i ∈ s, f i ≠ ⊤ ↔ ∀ i ∈ s, f i ≠ ⊤ := by simp
variable [LT α]
| Mathlib/Algebra/BigOperators/WithTop.lean | 31 | 33 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | @[simp]
theorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=
rfl
| Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 708 | 710 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | exact hf (congr_arg (fun x => coeff x p) h :)
theorem surjective (f : α → β) (hf : Surjective f) : Surjective (mapFun f : 𝕎 α → 𝕎 β) := fun x =>
⟨mk _ fun n => Classical.choose <| hf <| x.coeff n,
| Mathlib/RingTheory/WittVector/Basic.lean | 73 | 76 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex cones
In a `𝕜`-module `E`, we define a convex cone as a set `s` such that `a • x + b • y ∈ s` w... | Mathlib/Analysis/Convex/Cone/Basic.lean | 641 | 642 | |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... |
theorem inner_comp_rightAngleRotation (x y : E) : ⟪J x, J y⟫ = ⟪x, y⟫ :=
LinearIsometryEquiv.inner_map_map J x y
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 263 | 265 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Topology.Alge... | theorem continuous_algebraMap_iff_smul [ContinuousMul A] :
Continuous (algebraMap R A) ↔ Continuous fun p : R × A => p.1 • p.2 := by
refine ⟨fun h => ?_, fun h => have : ContinuousSMul R A := ⟨h⟩; continuous_algebraMap _ _⟩
simp only [Algebra.smul_def]
exact (h.comp continuous_fst).mul continuous_snd
| Mathlib/Topology/Algebra/Algebra.lean | 47 | 51 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Seminorm
import Mathlib.GroupTheory.GroupAction.Pointwise
/-!
# The Minkowski functional, normed field version
In this file we define `(eg... | refine le_antisymm ?_ (le_egauge_smul_left _ _ _)
rw [ENNReal.le_div_iff_mul_le (by simp [*]) (by simp)]
calc
egauge 𝕜 (c • s) x * ‖c‖ₑ = egauge 𝕜 (c • s) x / ‖c⁻¹‖ₑ := by
rw [enorm_inv (by simpa), div_eq_mul_inv, inv_inv]
_ ≤ egauge 𝕜 (c⁻¹ • c • s) x := le_egauge_smul_left _ _ _
_ = egauge �... | Mathlib/Analysis/Convex/EGauge.lean | 165 | 171 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | rw [← eq, mul_comm, mul_add]
· rw [coeff_expand hp.bot_lt, if_neg h, mul_zero]
| Mathlib/Algebra/Polynomial/Expand.lean | 224 | 226 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... |
/-- Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 662 | 663 |
/-
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... | sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
| Mathlib/Data/Set/Lattice.lean | 1,293 | 1,295 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Logic.Equiv.Defs
/-!
# Transferring `Traversable` instances along isomorphisms
This file allows to transfer `Trav... | protected theorem naturality (f : α → F β) (x : t' α) :
η (Equiv.traverse eqv f x) = Equiv.traverse eqv (@η _ ∘ f) x := by
| Mathlib/Control/Traversable/Equiv.lean | 129 | 130 |
/-
Copyright (c) 2017 Mario Carneiro. 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.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... |
See `opow_le_iff_le_log` for a variant assuming `x ≠ 0` rather than `c ≠ 0`. See also
| Mathlib/SetTheory/Ordinal/Exponential.lean | 343 | 344 |
/-
Copyright (c) 2020 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Order.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
import Batteries.WF
/-!
# Unbundled... | Mathlib/Order/RelClasses.lean | 611 | 611 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.Tilted
/-!
# Log-likelihood Ratio
The likelihood ratio between two measures `μ` and `ν` is their Radon-Nikodym derivative
`μ.rnDeri... | filter_upwards [hμν.ae_le (exp_llr μ ν), Measure.rnDeriv_pos hμν] with x hx_eq hx_pos
rw [hx_eq, if_neg hx_pos.ne']
lemma exp_llr_of_ac' (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] (hμν : ν ≪ μ) :
| Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | 52 | 55 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,288 | 1,294 | |
/-
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.Fin.Fin2
import Mathlib.Util.Notation3
import Mathlib.Tactic.TypeStar
/-!
# Alternate definition of `Vector` in terms of `Fin2`
This file provid... | theorem vectorAllP_cons (p : α → Prop) (x : α) (v : Vector3 α n) :
VectorAllP p (x :: v) ↔ p x ∧ VectorAllP p v :=
| Mathlib/Data/Vector3.lean | 236 | 237 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | @[ext]
| Mathlib/Algebra/Polynomial/Basic.lean | 714 | 714 |
/-
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.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 2,707 | 2,725 | |
/-
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.Order.Bounds.Basic
/-!
# Intervals in Lattices
In this file, we provide instances of lattice structures on intervals within lattices.
Some of them de... | refine ⟨fun h b hb ↦ ?_, fun h ↦ ⟨fun ⟨x, hx⟩ ↦ ?_⟩⟩
· obtain ⟨⟨c, hc₁⟩, hc⟩ := exists_isCompl (⟨b, hb⟩ : Iic a)
obtain ⟨hc₂, hc₃⟩ := Set.Iic.isCompl_iff.mp hc
| Mathlib/Order/LatticeIntervals.lean | 131 | 133 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | instance addLeftCancelSemigroup : AddLeftCancelSemigroup ℚ := by infer_instance
| Mathlib/Data/Rat/Defs.lean | 307 | 307 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.DiscreteQuotient
import Mathlib.Topology.Category.TopCa... | have hj : ∀ a, j a ∈ (Finset.univ.image j : Finset J) := by
intro a
simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply]
let fs : ∀ a : α, j0 ⟶ j a := fun a => (hj0 (hj a)).some
let gg : α → LocallyConstant (F.obj j0) (Fin 2) := fun a => (g a).comap (F.map (fs _)).hom
let ggg :... | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | 130 | 161 |
/-
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.Algebra.RestrictScalars
import Mathlib.Algebra.Lie.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.TensorPro... | · rintro - ⟨x, hx, m, hm, rfl⟩
exact ⟨1 ⊗ₜ x, tmul_mem_baseChange_of_mem 1 hx,
1 ⊗ₜ m, tmul_mem_baseChange_of_mem 1 hm, by simp⟩
· rintro - ⟨x, hx, m, hm, rfl⟩
revert m
apply Submodule.span_induction
(p := fun x' _ ↦ ∀ m' ∈ N.baseChange A, ⁅x', m'⁆ ∈ Submodule.span A s) (hx := hx)
·... | Mathlib/Algebra/Lie/BaseChange.lean | 218 | 251 |
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