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) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... |
@[simp]
theorem card_disjUnion (s t : Finset α) (h) : #(s.disjUnion t h) = #s + #t :=
Multiset.card_add _ _
| Mathlib/Data/Finset/Card.lean | 468 | 472 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | calc
_ = ∑ i ∈ s, w i ^ (1 - p⁻¹) * (w i ^ p⁻¹ * f i) := ?_
_ ≤ (∑ i ∈ s, (w i ^ (1 - p⁻¹)) ^ (1 - p⁻¹)⁻¹) ^ (1 / (1 - p⁻¹)⁻¹) *
(∑ i ∈ s, (w i ^ p⁻¹ * f i) ^ p) ^ (1 / p) :=
inner_le_Lp_mul_Lq _ _ _ (.symm <| Real.holderConjugate_iff.mpr ⟨hp, by simp⟩)
_ = _ := ?_
· congr with i
r... | Mathlib/Analysis/MeanInequalities.lean | 525 | 537 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | | iUnion f hfd hfm ihf =>
simp only [← restrict_apply (hfm _), ← restrict_apply (MeasurableSet.iUnion hfm)] at ihf ⊢
simp only [measure_iUnion hfd hfm, ihf]
| Mathlib/MeasureTheory/Measure/Restrict.lean | 437 | 439 |
/-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Metric
/-!
#... | classical
-- The case `(G □ H).edist x y = ⊤` is used twice, so better to factor it out.
have top_case : (G □ H).edist x y = ⊤ ↔ G.edist x.1 y.1 = ⊤ ∨ H.edist x.2 y.2 = ⊤ := by
simp_rw [← not_ne_iff, edist_ne_top_iff_reachable, boxProd_reachable, not_and_or]
by_cases h : (G □ H).edist x y = ⊤
| Mathlib/Combinatorics/SimpleGraph/Prod.lean | 244 | 248 |
/-
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.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | @[reassoc (attr := simp)] lemma w₁₃_τ₂ : S.v₁₂.τ₂ ≫ S.v₂₃.τ₂ = 0 := by
rw [← comp_τ₂, S.w₁₃, zero_τ₂]
| Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 126 | 127 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | theorem ae_eq_set_union {s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t') :
(s ∪ s' : Set α) =ᵐ[μ] (t ∪ t' : Set α) :=
| Mathlib/MeasureTheory/OuterMeasure/AE.lean | 184 | 185 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | variable [Field α] {a b c p : α}
| Mathlib/Algebra/ModEq.lean | 287 | 287 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Contractions
Given modules $M, N$ over a commutative ring $R$, t... | provides this equivalence in return for a basis of `M`. -/
-- We manually create simp-lemmas because `@[simps]` generates a malformed lemma
noncomputable def dualTensorHomEquivOfBasis : Module.Dual R M ⊗[R] N ≃ₗ[R] M →ₗ[R] N :=
LinearEquiv.ofLinear (dualTensorHom R M N)
(∑ i, TensorProduct.mk R _ N (b.dualBasis i... | Mathlib/LinearAlgebra/Contraction.lean | 133 | 140 |
/-
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.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | gcongr
· apply le_max_left
· apply le_max_right
_ = 2 * maxVar X Y * max (dist x z) (dist y t) := by
rw [dist_comm t y]
ring
_ = 2 * maxVar X Y * dist (x, y) (z, t) := rfl
/-- Candidates are Lipschitz -/
private theorem candidates_lipschitz (fA : f ∈ candidates X Y) :
Lipschit... | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 180 | 195 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | (∫ x in a..b, cos x ^ (n + 2)) = ∫ x in a..b, cos x ^ (n + 1) * cos x := by
simp only [_root_.pow_succ]
_ = C + (n + 1) * ∫ x in a..b, sin x ^ 2 * cos x ^ n := by simp [C, H, h, sq, -neg_add_rev]
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 705 | 707 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
import Mathlib.Logic.Function.ULift
/-!
# The Yoneda emb... |
lemma isIso_of_yoneda_map_bijective {X Y : C} (f : X ⟶ Y)
(hf : ∀ (T : C), Function.Bijective (fun (x : T ⟶ X) => x ≫ f)) :
IsIso f := by
obtain ⟨g, hg : g ≫ f = 𝟙 Y⟩ := (hf Y).2 (𝟙 Y)
exact ⟨g, (hf _).1 (by aesop_cat), hg⟩
| Mathlib/CategoryTheory/Yoneda.lean | 632 | 637 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.Algebra.Group.ULift
import Mathlib.Data.Int.ModEq
import M... | lemma charP_iff_prime_eq_zero [Nontrivial R] {p : ℕ} (hp : p.Prime) :
CharP R p ↔ (p : R) = 0 :=
⟨fun _ => cast_eq_zero R p,
fun hp0 => (ringChar_of_prime_eq_zero hp hp0) ▸ inferInstance⟩
| Mathlib/Algebra/CharP/Basic.lean | 98 | 101 |
/-
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... |
@[simp]
| Mathlib/Analysis/Convex/Between.lean | 378 | 379 |
/-
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.Combinatorics.SimpleGraph.Prod
import Mathlib.Data.Fin.SuccPred
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Relation
import Mathlib.Tact... | theorem pathGraph_two_eq_top : pathGraph 2 = ⊤ := by
ext u v
fin_cases u <;> fin_cases v <;> simp [pathGraph, ← Fin.coe_covBy_iff, covBy_iff_add_one_eq]
| Mathlib/Combinatorics/SimpleGraph/Hasse.lean | 108 | 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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Ma... | theorem ker_single (i : ι) : ker (single R φ i) = ⊥ :=
ker_eq_bot_of_injective <| Pi.single_injective _ _
theorem proj_comp_single (i j : ι) : (proj i).comp (single R φ j) = diag j i := by
| Mathlib/LinearAlgebra/Pi.lean | 317 | 320 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... |
/-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on the
interior, then `f` is strictly convex on `D`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly positive, except at at most one point. -/
theore... | Mathlib/Analysis/Convex/Deriv.lean | 259 | 273 |
/-
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.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import ... | (G.map f).EdgeDisjointTriangles := by
| Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean | 62 | 62 |
/-
Copyright (c) 2022 Yaël Dillies. 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.Algebra.CharP.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.G... | map_prod_eq_map_prod s t hsA htA hs ht h := by
rw [Pi.inv_def, prod_map_inv, prod_map_inv, hf.map_prod_eq_map_prod hsA htA hs ht h]
@[to_additive] lemma IsMulFreimanHom.div {β : Type*} [DivisionCommMonoid β] {B₁ B₂ : Set β}
{f₁ f₂ : α → β} (h₁ : IsMulFreimanHom n A B₁ f₁) (h₂ : IsMulFreimanHom n A B₂ f₂) :
... | Mathlib/Combinatorics/Additive/FreimanHom.lean | 332 | 350 |
/-
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 | 2,154 | 2,154 | |
/-
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... | Mathlib/Logic/Equiv/PartialEquiv.lean | 1,156 | 1,158 | |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | Mathlib/Data/Fin/Tuple/Basic.lean | 1,192 | 1,197 | |
/-
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 | 834 | 849 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | Mathlib/Data/Nat/Bitwise.lean | 395 | 396 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | exact
match f a, g b, hfg h₁ with
| _, _, Option.Rel.none => rel_filterMap (@hfg) h₂
| _, _, Option.Rel.some h => Forall₂.cons h (rel_filterMap (@hfg) h₂)
| Mathlib/Data/List/Forall2.lean | 262 | 266 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... | [T2Space Y] [Infinite X] :
C(OnePoint X, Y) ≃ { f : X → Y // ∃ L, Tendsto (fun x : X ↦ f x) cofinite (𝓝 L) } where
toFun f := ⟨(f ·), ⟨f ∞, continuous_iff_from_discrete _ |>.mp (map_continuous f)⟩⟩
invFun f :=
| Mathlib/Topology/Compactification/OnePoint.lean | 412 | 415 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | theorem fib_succ_eq_succ_sum (n : ℕ) : fib (n + 1) = (∑ k ∈ Finset.range n, fib k) + 1 := by
induction' n with n ih
· simp
· calc
fib (n + 2) = fib n + fib (n + 1) := fib_add_two
_ = (fib n + ∑ k ∈ Finset.range n, fib k) + 1 := by rw [ih, add_assoc]
_ = (∑ k ∈ Finset.range (n + 1), fib k) + 1 :=... | Mathlib/Data/Nat/Fib/Basic.lean | 249 | 258 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.RingTheory.Ideal.Maps
/-!
# Ideals in product rings
For commutative rings `R` and `S` and ideals `I ≤ R`, `J ≤ S`, we define `Ideal.prod I J` as the
pr... | `S`. -/
def idealProdEquiv : Ideal (R × S) ≃o Ideal R × Ideal S where
toFun I := ⟨map (RingHom.fst R S) I, map (RingHom.snd R S) I⟩
invFun I := prod I.1 I.2
left_inv I := (ideal_prod_eq I).symm
right_inv := fun ⟨I, J⟩ => by simp
map_rel_iff' {I J} := by
| Mathlib/RingTheory/Ideal/Prod.lean | 72 | 78 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Notation
/-!
# Positive & negative parts
Mathematical structures posse... | oneLePart_div_leOnePart, leOnePart_eq_inv_inf_one, inv_inv]
| Mathlib/Algebra/Order/Group/PosPart.lean | 168 | 169 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | induction' n with n hn IH generalizing l
· simp
· rcases l with - | ⟨hd, tl⟩
· simp
· simp [hn]
| Mathlib/Data/List/Rotate.lean | 327 | 332 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Fourie... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 232 | 237 | |
/-
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.... | curry_apply, coe_comp, coe_restrictScalars, coe_inr, Function.comp_apply, prodMap_apply,
map_zero, prodMapLinear_apply, zero_prodMap_dualTensorHom, trace_eq_contract_apply,
contractLeft_apply, coe_snd, coprod_apply, id_coe, id_eq, zero_add, e]
variable {R M N P}
| Mathlib/LinearAlgebra/Trace.lean | 200 | 204 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | simp
theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h := by
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 951 | 954 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Patrick Stevens
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Tactic.Common
import Mathlib.Algebra.Field.Defs
... | obtain ⟨k, rfl⟩ := hnm
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Nat.mul_div_cancel_left _ <| zero_lt_of_ne_zero this, mul_comm n,
cast_mul, mul_div_cancel_right₀ _ hn]
| Mathlib/Data/Nat/Cast/Field.lean | 29 | 33 |
/-
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... | simp_rw [← iInf_lowerCentralSeries_eq_posFittingComp, hl₁, hl₂, hl₃,
LieSubmodule.lowerCentralSeries_map_eq_lcs, ← LieSubmodule.lcs_sup, lowerCentralSeries,
h.eq_top]
lemma genWeightSpace_genWeightSpaceOf_map_incl (x : L) (χ : L → R) :
(genWeightSpace (genWeightSpaceOf M (χ x) x) χ).map (genWeightSpaceOf... | Mathlib/Algebra/Lie/Weights/Basic.lean | 575 | 606 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | /-- The inner product of a nonzero vector with a nonzero multiple of
itself, divided by the product of their norms, has absolute value
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 641 | 642 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | le_total := by
intro a b
transfer_rw
apply le_total
toDecidableLT := by infer_instance
| Mathlib/Data/Num/Lemmas.lean | 559 | 563 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Set.Lattice.Image
/-!
# Naturals pairing function
Th... | theorem unpair_pair (a b : ℕ) : unpair (pair a b) = (a, b) := by
dsimp only [pair]; split_ifs with h
| Mathlib/Data/Nat/Pairing.lean | 57 | 58 |
/-
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 | 2,150 | 2,151 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Submo... |
end IsUnit
namespace MonoidHom
open Submonoid
/-- Let `s` be a subset of a monoid `M` such that the closure of `s` is the whole monoid.
| Mathlib/Algebra/Group/Submonoid/Basic.lean | 356 | 363 |
/-
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... | refine x.induction_on ?_ ?_ ?_
· simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
· intro a₁ l₁
refine y.induction_on ?_ ?_ ?_
· simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
· intro a₂ l₂
refine z.induction_on ?_ ?_ ?_
· simp only [LinearMap.map_zero, add_ze... | Mathlib/Algebra/Lie/BaseChange.lean | 85 | 107 |
/-
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... | apply le_trans (dist_triangle4 a d e f)
apply add_le_add_right (add_le_add_right _ (dist d e)) (dist e f)
exact dist_triangle4 a b c d
theorem swap_dist : Function.swap (@dist α _) = dist := by funext x y; exact dist_comm _ _
theorem abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y :=
abs_sub_le... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 216 | 225 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
/-... |
end EssSurj
variable {J C D : Type*} [Category J] [Category C] [Category D]
| Mathlib/CategoryTheory/EssentialImage.lean | 169 | 172 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Sups
import Mathlib.Logic.Function.Iterate
/-!
# Shadows
This file defines shad... | aesop
/-- `s ∈ ∂ 𝒜` iff `s` is exactly one element less than something from `𝒜`.
See also `Finset.mem_shadow_iff_exists_sdiff`. -/
| Mathlib/Combinatorics/SetFamily/Shadow.lean | 106 | 110 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
/-!
# Filtered categories
A category is filtered if every finite diagram admits a cocone.
We give a... | to `s`, making the resulting squares commute.
-/
theorem tulip {j₁ j₂ j₃ k₁ k₂ l : C} (f₁ : j₁ ⟶ k₁) (f₂ : j₂ ⟶ k₁) (f₃ : j₂ ⟶ k₂) (f₄ : j₃ ⟶ k₂)
(g₁ : j₁ ⟶ l) (g₂ : j₃ ⟶ l) :
∃ (s : C) (α : k₁ ⟶ s) (β : l ⟶ s) (γ : k₂ ⟶ s),
f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = f₃ ≫ γ ∧ f₄ ≫ γ = g₂ ≫ β := by
obtain ⟨l', k₁l, k₂l,... | Mathlib/CategoryTheory/Filtered/Basic.lean | 466 | 472 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | Mathlib/Data/Finset/Card.lean | 865 | 883 | |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | rw [finSepDegree_eq_of_equiv _ _ _ (botEquiv F E), finSepDegree_self]
| Mathlib/FieldTheory/SeparableDegree.lean | 179 | 180 |
/-
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... | hy ▸ ⟨fa x, h x⟩
theorem surjOn_image (h : Semiconj f fa fb) (ha : SurjOn fa s t) : SurjOn fb (f '' s) (f '' t) := by
| Mathlib/Data/Set/Function.lean | 1,098 | 1,100 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | ext x y; exact H x y
lemma PartialOrder.ext {A B : PartialOrder α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by ext x y; exact H x y
lemma PartialOrder.ext_lt {A B : PartialOrder α} (H : ∀ x y : α, (haveI := A; x < y) ↔ x < y) :
A = B := by ext x y; rw [le_iff_lt_or_eq, @le_iff_lt_or_eq _ A, H... | Mathlib/Order/Basic.lean | 612 | 638 |
/-
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.Order.SuccPred.Archimedean
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Successor and predecessor limits
We define the pre... | simp [h.le_iff_forall_le]
| Mathlib/Order/SuccPred/Limit.lean | 283 | 284 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | ((s ×ˢ t).filter fun x : α × β => p x.1 ∧ q x.2) = s.filter p ×ˢ t.filter q := by
ext ⟨a, b⟩
simp [mem_filter, mem_product, decide_eq_true_eq, and_comm, and_left_comm, and_assoc]
theorem filter_product_left (p : α → Prop) [DecidablePred p] :
| Mathlib/Data/Finset/Prod.lean | 147 | 151 |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.Units.Defs
import Mathlib.Algebra.BigOperators.Group.List.Basic
import Mathlib.Algebr... |
section Mem
variable {m : α}
| Mathlib/Algebra/FreeMonoid/Basic.lean | 199 | 201 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | {g : α → E} (hg : Measurable g) (f : SimpleFunc α E) :
Measurable (g + (f : α → E)) := by
classical
induction f using SimpleFunc.induction with
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,230 | 1,233 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | -- Porting note: the explicit motive required because `simp` does not apply `Part.get_eq_of_mem`
refine Eq.rec (motive := fun x e =>
Sum.casesOn (motive := fun y => (f a).get (dom_of_mem_fix h) = y → C a) x ?_ ?_
(Eq.trans (Part.get_eq_of_mem fa (dom_of_mem_fix h)) e) = _) ?_
(Part.get_eq_of_mem fa ... | Mathlib/Data/PFun.lean | 338 | 344 |
/-
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.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.Typeclasse... | theorem measure_Ioi_pos [NoMaxOrder X] (a : X) : 0 < μ (Ioi a) :=
isOpen_Ioi.measure_pos μ nonempty_Ioi
theorem measure_Iio_pos [NoMinOrder X] (a : X) : 0 < μ (Iio a) :=
isOpen_Iio.measure_pos μ nonempty_Iio
theorem measure_Ioo_pos [DenselyOrdered X] {a b : X} : 0 < μ (Ioo a b) ↔ a < b :=
| Mathlib/MeasureTheory/Measure/OpenPos.lean | 156 | 162 |
/-
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.Group.TypeTags.Finite
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleMod... |
instance [AddGroup G] : IsSemisimpleRing (AddMonoidAlgebra k G) :=
haveI : NeZero (Fintype.card (Multiplicative G) : k) := by
rwa [Fintype.card_congr Multiplicative.toAdd]
| Mathlib/RepresentationTheory/Maschke.lean | 166 | 169 |
/-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Ralf Stephan
-/
import Mathlib.Data.Nat.Prime.Nth
import Mathlib.Data.Nat.Totient
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Order.Filter.AtTopBot.Basic
/-... | convert surjective_primeCounting'
ext
exact primeCounting_sub_one _
open Filter
theorem tendsto_primeCounting' : Tendsto π' atTop atTop := by
apply tendsto_atTop_atTop_of_monotone' monotone_primeCounting'
simp [Set.range_eq_univ.mpr surjective_primeCounting']
theorem tensto_primeCounting : Tendsto π atTop ... | Mathlib/NumberTheory/PrimeCounting.lean | 83 | 102 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib... | simp only [← of_symm_smul_of_eq_mul]
exact HahnModule.smul_add x y z
right_distrib := fun x y z => by
simp only [← of_symm_smul_of_eq_mul]
refine HahnModule.add_smul ?_
simp only [smul_eq_mul]
exact add_mul }
| Mathlib/RingTheory/HahnSeries/Multiplication.lean | 430 | 436 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 1,473 | 1,477 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.AlgebraicTopology.SimplicialObject.Basic
import Mathlib.CategoryTheory.Abelian.Basic
/-!
## Moore compl... | ← Fin.castSucc_zero,
← X.δ_comp_δ (Fin.zero_le (0 : Fin (n + 3))),
← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin (n + 3)) (by simp)),
Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]
/-- The normalized Moore complex functor, on objects.
-/
@[simp... | Mathlib/AlgebraicTopology/MooreComplex.lean | 100 | 111 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Equiv
import Mathlib.... | (@decode (α × β) _ n : Option (α × β))
= (decode n.unpair.1).bind fun a => (decode n.unpair.2).map <| Prod.mk a := by
simp only [decode_ofEquiv, Equiv.symm_symm, decode_sigma_val]
cases (decode n.unpair.1 : Option α) <;> cases (decode n.unpair.2 : Option β)
<;> rfl
@[simp]
theorem encode_prod_val (a b)... | Mathlib/Logic/Encodable/Basic.lean | 325 | 334 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.RingTheory.Finiteness.Nakayama
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheor... | rintro _ ⟨m, hm, rfl⟩
refine Submodule.smul_induction_on hm (fun r hr m' hm' => ?_) (fun x y hx hy => ?_)
· rw [add_comm, ← monomial_smul_single]
exact F'.smul_mem
| Mathlib/RingTheory/Filtration.lean | 318 | 321 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | intro ε hε
obtain ⟨ε', hε'⟩ := exists_rat_btwn hε
norm_cast at hε'
obtain ⟨N, hN⟩ := hq ε' hε'.1
exists N
intro i hi
have h := hN i hi
change norm (f i - q) < ε
refine lt_trans ?_ hε'.2
dsimp [norm]
exact mod_cast h
theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 894 | 909 |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | simp only [cauchy_map_iff, hp, true_and]
intro u hu
rw [mem_map]
filter_upwards [hf u hu] with p hp using hp x hx
/-- If a sequence of functions is uniformly Cauchy on a set, then the values at each point form
| Mathlib/Topology/UniformSpace/UniformConvergence.lean | 494 | 499 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,
LinearEquiv.symm_apply_apply, hg]
simp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp]
have Ce : Continuous e := (e : E →ₗ[ℝ] ι → ℝ).continuous_of_finiteDimensional
have Cg : Continuous g := LinearMap.continuous_of_finiteDimensi... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 242 | 267 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-20")
| Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 118 | 120 | |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 156 | 158 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.IntegralCompProd
import Mathlib.Probability.Kernel.Disintegration.StandardBorel
/-!
# Lebesgue and Bochner integrals of con... |
section Integral
variable {ρ : Measure (β × Ω)} [IsFiniteMeasure ρ]
| Mathlib/Probability/Kernel/Disintegration/Integral.lean | 182 | 185 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... | Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 170 | 177 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... | @[simps! apply]
def mapRange.linearMap (f : ∀ i, β₁ i →ₗ[R] β₂ i) : (Π₀ i, β₁ i) →ₗ[R] Π₀ i, β₂ i :=
{ mapRange.addMonoidHom fun i => (f i).toAddMonoidHom with
toFun := mapRange (fun i x => f i x) fun i => (f i).map_zero
| Mathlib/LinearAlgebra/DFinsupp.lean | 206 | 209 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... | ext : 1
exact (torsionBySet_span_singleton_eq _).symm
exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
theorem supIndep_torsionBy (hq : (S : Set ι).Pairwise <| (IsCoprime on q)) :
S.SupIndep fun i => torsionBy R M <| q i := by
convert supIndep_torsionBySet_ideal (M := M)... | Mathlib/Algebra/Module/Torsion.lean | 439 | 447 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | theorem liftProp_of_locally_liftPropOn (h : ∀ x, ∃ u, IsOpen u ∧ x ∈ u ∧ LiftPropOn P g u) :
LiftProp P g := by
rw [← liftPropOn_univ]
refine hG.liftPropOn_of_locally_liftPropOn fun x _ ↦ ?_
simp [h x]
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 364 | 369 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | rw [← Finsupp.log_prod, ← Nat.cast_finsuppProd, Nat.factorization_prod_pow_eq_self hn]
intro p hp
rw [pow_eq_zero (Nat.cast_eq_zero.1 hp), Nat.factorization_zero_right]
theorem tendsto_pow_log_div_mul_add_atTop (a b : ℝ) (n : ℕ) (ha : a ≠ 0) :
Tendsto (fun x => log x ^ n / (a * x + b)) atTop (𝓝 0) :=
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 385 | 390 |
/-
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... | simp only [AddMonoidHom.coe_comp, Finsupp.mapRange_single, Finsupp.mapDomain.addMonoidHom_apply,
Finsupp.singleAddHom_apply, eq_self_iff_true, Function.comp_apply, Finsupp.mapDomain_single,
Finsupp.mapRange.addMonoidHom_apply]
/-- When `g` preserves addition, `mapRange` and `mapDomain` commute. -/
theorem ma... | Mathlib/Data/Finsupp/Basic.lean | 559 | 570 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.PolynomialGaloisGroup
/-!
# Galois group of cyclotomic extensions
In this file, we ... | apply_fun Units.val at hfg
simp only [IsPrimitiveRoot.coe_autToPow_apply] at hfg
generalize_proofs hn₀ hf' hg' at hfg
have hf := hf'.choose_spec
have hg := hg'.choose_spec
generalize_proofs hζ at hf hg
suffices f (hμ.toRootsOfUnity : Lˣ) = g (hμ.toRootsOfUnity : Lˣ) by
apply AlgEquiv.coe_algHom_inject... | Mathlib/NumberTheory/Cyclotomic/Gal.lean | 55 | 76 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
/-! The unit isomorphism of the Dold-Kan equivalence
In order to con... | PInfty.f n ≫ X.map i.op = 0 := by
induction' Δ' using SimplexCategory.rec with m
obtain ⟨k, hk⟩ := Nat.exists_eq_add_of_lt (len_lt_of_mono i fun h => by
rw [← h] at h₁
exact h₁ rfl)
simp only [len_mk] at hk
rcases k with _|k
· change n = m + 1 at hk
subst hk
obtain ⟨j, rfl⟩ := eq_δ... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 38 | 78 |
/-
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... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,410 | 1,422 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | @[simp]
theorem boundedBy_eq_self (m : OuterMeasure α) : boundedBy m = m :=
ext fun _ => boundedBy_eq _ measure_empty (fun _ ht => measure_mono ht) measure_iUnion_le
| Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 270 | 272 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
/-!
# Contents
In this file... | Mathlib/MeasureTheory/Measure/Content.lean | 438 | 448 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... | Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 212 | 221 | |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | theorem arccos_neg (x : ℝ) : arccos (-x) = π - arccos x := by
rw [← add_halves π, arccos, arcsin_neg, arccos, add_sub_assoc, sub_sub_self, sub_neg_eq_add]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 353 | 354 |
/-
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.ε'_succ p q hpq
| Mathlib/Algebra/Homology/ComplexShapeSigns.lean | 138 | 139 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
/-!
# Split si... | cofan_inj_πSummand_eq_zero]
rw [ne_comm]
change ¬(A.epiComp (SimplexCategory.σ i).op).EqId
rw [IndexSet.eqId_iff_len_eq]
have h := SimplexCategory.len_le_of_epi (inferInstance : Epi A.e)
dsimp at h ⊢
omega
/-- If a simplicial object `X` in an additive category is split,
then `PInfty` vanishes on all th... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 73 | 85 |
/-
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.Nonneg.Field
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Tactic.Positivity.Basic
/-!
# Some exiled lemmas about cas... | theorem cast_zpow_of_ne_zero {K} [DivisionSemiring K] (q : ℚ≥0) (z : ℤ) (hq : (q.num : K) ≠ 0) :
NNRat.cast (q ^ z) = (NNRat.cast q : K) ^ z := by
obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg
· simp
· simp_rw [zpow_neg, zpow_natCast, ← inv_pow, NNRat.cast_pow]
congr
rw [cast_inv_of_ne_zero hq]
| Mathlib/Data/Rat/Cast/Lemmas.lean | 69 | 75 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Module.Equiv
/-! # Infinite sums in top... |
protected theorem ContinuousLinearEquiv.summable {f : ι → M} (e : M ≃SL[σ] M₂) :
(Summable fun b : ι ↦ e (f b)) ↔ Summable f :=
⟨fun hf ↦ (e.hasSum.1 hf.hasSum).summable, (e : M →SL[σ] M₂).summable⟩
| Mathlib/Topology/Algebra/InfiniteSum/Module.lean | 150 | 153 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Constructions
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
/-! ## Equivalence of smoothne... |
theorem ContDiffWithinAt.comp_contMDiffWithinAt {g : F → F'} {f : M → F} {s : Set M} {t : Set F}
{x : M} (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContMDiffWithinAt I 𝓘(𝕜, F) n f s x)
| Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 63 | 65 |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... |
@[to_additive]
theorem mul_rotate' (a b c : G) : a * (b * c) = b * (c * a) := by
| Mathlib/Algebra/Group/Basic.lean | 128 | 130 |
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | refine ⟨fun b ↦ ⨆ i, (hf i).mk (f i) b, .iSup (fun i ↦ (hf i).measurable_mk), ?_⟩
filter_upwards [ae_all_iff.2 (fun i ↦ (hf i).ae_eq_mk)] with b hb using by simp [hb]
@[measurability, fun_prop]
protected theorem Measurable.iInf {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
Measurable fun b =... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 782 | 787 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | have b_ne_zero : b ≠ 0 := by linarith
have b_ne_minus_one : b ≠ -1 := by linarith
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 119 | 120 |
/-
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... | on_goal 1 => refine rpow_le_rpow ?_ hxy (neg_nonneg.2 hz)
all_goals positivity
theorem rpow_lt_rpow_iff (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z < y ^ z ↔ x < y :=
⟨lt_imp_lt_of_le_imp_le fun h => rpow_le_rpow hy h (le_of_lt hz), fun h => rpow_lt_rpow hx h hz⟩
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 543 | 548 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... |
@[simp]
theorem head?_cyclicPermutations (l : List α) : (cyclicPermutations l).head? = l := by
rw [head?_eq_head, head_cyclicPermutations]
| Mathlib/Data/List/Rotate.lean | 537 | 541 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | hf
namespace Computable
| Mathlib/Computability/Partrec.lean | 257 | 260 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,826 | 1,832 | |
/-
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.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic
/-!
# Half braidings and the Drinfeld center of a monoidal c... |
instance : (forget C).ReflectsIsomorphisms where
reflects f i := by dsimp at i; change IsIso (isoMk f).hom; infer_instance
| Mathlib/CategoryTheory/Monoidal/Center.lean | 323 | 325 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | add_comm := by transfer
instance commMonoid : CommMonoid PosNum where
mul := (· * ·)
one := (1 : PosNum)
npow := @npowRec PosNum ⟨1⟩ ⟨(· * ·)⟩
mul_assoc := by transfer
one_mul := by transfer
mul_one := by transfer
mul_comm := by transfer
| Mathlib/Data/Num/Lemmas.lean | 526 | 536 |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... | e.cast hu hv = e' ↔ HEq e e' := by
rw [Hom.cast_eq_cast]
exact _root_.cast_eq_iff_heq
| Mathlib/Combinatorics/Quiver/Cast.lean | 57 | 60 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Opposite
import Mathlib.Alg... | f.comp (fst M N) * g.comp (snd M N)
@[to_additive (attr := simp)]
| Mathlib/Algebra/Group/Prod.lean | 447 | 449 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | refine ⟨x.denom * y.denom, mul_ne_zero (denom_ne_zero x) (denom_ne_zero y), ?_⟩
rw [RingHom.map_mul, RingHom.map_mul, ← div_mul_div_comm, num_div_denom, num_div_denom]
theorem denom_mul_dvd (x y : RatFunc K) : denom (x * y) ∣ denom x * denom y := by
| Mathlib/FieldTheory/RatFunc/Basic.lean | 956 | 959 |
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Complex.PhragmenLindelof
/-!
# Hadamard three-lines Theorem
In this file we present a ... | · intro h
obtain ⟨z, hz₁, hz₂⟩ := h
use ↑l + z * (↑u - ↑l)
simp [hz₁, hz₂]
· intro h
obtain ⟨z, hz₁, hz₂⟩ := h
use ((z - l) / (u - l))
constructor
· norm_cast
rw [Complex.div_re, Complex.normSq_ofReal, Complex.ofReal_re]
simp[hz₁]
· rw [div_mul_com... | Mathlib/Analysis/Complex/Hadamard.lean | 370 | 416 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen,
Kim Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimensi... | ⇑(basisOfTopLeSpanOfCardEqFinrank b le_span card_eq) = b :=
Basis.coe_mk _ _
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 196 | 198 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | theorem chromaticNumber [Fintype ι] (f : ∀ (i : ι), V i) :
(completeMultipartiteGraph V).chromaticNumber = Fintype.card ι := by
apply le_antisymm (colorable V).chromaticNumber_le
by_contra! h
exact not_cliqueFree_of_le_card V f le_rfl <| cliqueFree_of_chromaticNumber_lt h
theorem colorable_of_cliqueFree (f :... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 467 | 473 |
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