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/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | lemma inl_descCochain :
(inl φ).comp (descCochain φ α β h) (by omega) = α := by
simp [descCochain]
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 302 | 305 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,006 | 1,007 | |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara, Rida Hamadani
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.ENat.Lattice
/-!
# Graph metric
This module defines the `SimpleGraph.edi... | lemma exists_walk_of_edist_ne_top (h : G.edist u v ≠ ⊤) :
∃ p : G.Walk u v, p.length = G.edist u v :=
(reachable_of_edist_ne_top h).exists_walk_length_eq_edist
| Mathlib/Combinatorics/SimpleGraph/Metric.lean | 90 | 92 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... |
@[simp]
theorem seq_some {α β} {a : α} {f : α → β} : some f <*> some a = some (f a) :=
rfl
| Mathlib/Data/Option/Basic.lean | 195 | 198 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | /-- Local syntax for composition of relations. -/
-- TODO: this could be replaced with `local infixr:90 " ∘ " => Rel.comp`.
local infixr:90 " • " => Rel.comp
| Mathlib/Data/Rel.lean | 86 | 88 |
/-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Data.FunLike... | theorem mapsTo_ball (x : α) (r' : ℝ) :
MapsTo (f : α → β) (Metric.ball x r') (Metric.ball (f x) (ratio f * r')) :=
fun y hy => (dist_eq f y x).trans_lt <| (mul_lt_mul_left <| NNReal.coe_pos.2 <| ratio_pos f).2 hy
| Mathlib/Topology/MetricSpace/Dilation.lean | 460 | 463 |
/-
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.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 657 | 658 | |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | theorem totalDegree_multiset_prod (s : Multiset (MvPolynomial σ R)) :
s.prod.totalDegree ≤ (s.map MvPolynomial.totalDegree).sum := by
refine Quotient.inductionOn s fun l => ?_
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 455 | 457 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,702 | 2,704 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,613 | 1,616 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Star.Module
import Mathlib.Algebra.Star.NonUn... | theorem coe_comap (S : StarSubalgebra R B) (f : A →⋆ₐ[R] B) :
(S.comap f : Set A) = f ⁻¹' (S : Set B) :=
rfl
| Mathlib/Algebra/Star/Subalgebra.lean | 280 | 282 |
/-
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... |
theorem ball_add_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
ball a ε + ball b δ = ball (a + b) (ε + δ) := by
rw [ball_add, thickening_ball hε hδ b, Metric.vadd_ball, vadd_eq_add]
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 320 | 324 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | | ⟨aI, haI, hI⟩, J =>
⟨aI, haI, fun b hb => by
rcases mem_inf.mp hb with ⟨hbI, _⟩
exact hI b hbI⟩
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 425 | 429 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 853 | 859 | |
/-
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.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we have `image_le_ker... | Mathlib/Algebra/Homology/ImageToKernel.lean | 340 | 344 | |
/-
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... | | pos p => congr_arg pos p.bit1_of_bit1
| Mathlib/Data/Num/Lemmas.lean | 198 | 199 |
/-
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... | simp only [← Set.Ioo_insert_right ha, Set.insert_subset_iff, ho, true_and]
· simp [nhds_eq_pure.2 ho', ho, ho']
open List Set in
theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
TFAE [a ∈ closure s,
a ∈ closure (s ∩ Iic a),
(s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t, t ⊆... | Mathlib/SetTheory/Ordinal/Topology.lean | 86 | 124 |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
/-!
# Diagonal matrices
This file co... | theorem transpose_mul_self_isDiag_iff_hasOrthogonalCols [Fintype m] [Mul α] [AddCommMonoid α]
{A : Matrix m n α} : (Aᵀ * A).IsDiag ↔ A.HasOrthogonalCols :=
Iff.rfl
end Matrix
| Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 169 | 179 |
/-
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.... | classical
rw [Multiset.mem_toFinset, mem_aroots]
exact ⟨ne_of_apply_ne _ h, map_rootOfSplits _ (SplittingField.splits f) (ne_of_apply_ne _ h)⟩
/-- A polynomial has zero separable degree if and only if it is constant. -/
| Mathlib/FieldTheory/SeparableDegree.lean | 333 | 337 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | @[to_additive]
theorem finprod_mem_eq_one_of_infinite {f : α → M} {s : Set α} (hs : (s ∩ mulSupport f).Infinite) :
∏ᶠ i ∈ s, f i = 1 := by
rw [finprod_mem_def]
apply finprod_of_infinite_mulSupport
rwa [← mulSupport_mulIndicator] at hs
@[to_additive]
theorem finprod_mem_eq_one_of_forall_eq_one {f : α → M} {s ... | Mathlib/Algebra/BigOperators/Finprod.lean | 465 | 476 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | (indicator (closure E) (fun _ => (1 : ℝ≥0∞)) x).toNNReal
by refine (congr_fun (comp_indicator_const 1 ENNReal.toNNReal toNNReal_zero) x).symm]
refine Tendsto.comp (tendsto_toNNReal ?_) (key x)
by_cases x_mem : x ∈ closure E <;> simp [x_mem]
| Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 227 | 231 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | by_cases hq : q.Monic
· rcases subsingleton_or_nontrivial R with hR | hR
· simp only [eq_iff_true_of_subsingleton]
| Mathlib/Algebra/Polynomial/Div.lean | 670 | 672 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... | lemma mem_smoothNumbers_iff_forall_le {n m : ℕ} :
m ∈ smoothNumbers n ↔ m ≠ 0 ∧ ∀ p ≤ m, p.Prime → p ∣ m → p < n := by
simp only [smoothNumbers_eq_factoredNumbers, mem_factoredNumbers_iff_forall_le, Finset.mem_range]
| Mathlib/NumberTheory/SmoothNumbers.lean | 303 | 305 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | rw [sub_eq_add_neg, sub_eq_add_neg, reflect_add, reflect_neg]
| Mathlib/Algebra/Polynomial/Reverse.lean | 363 | 364 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... |
theorem disjSups_inter_subset_right : s ○ (t₁ ∩ t₂) ⊆ s ○ t₁ ∩ s ○ t₂ := by
| Mathlib/Data/Finset/Sups.lean | 463 | 464 |
/-
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
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | Theorems in this section work both for real and complex differentiable functions. We use assumptions
`[NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to
achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion
of a `Convex` set. -/
section... | Mathlib/Analysis/Calculus/MeanValue.lean | 401 | 406 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.CompHausLike.Limits
import Mathlib.Topology.Category.LightProfinite.Basic
/-!
# Explicit limits and colimits
This file applies ... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 86 | 94 | |
/-
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 ... | toNonempty : Nonempty α
attribute [instance 50] ConnectedSpace.toNonempty -- see Note [lower instance priority]
-- see Note [lower instance priority]
| Mathlib/Topology/Connected/Basic.lean | 620 | 624 |
/-
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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | rw [H]
exact cof_lsub_le_lift f
theorem cof_iSup_le {ι} {f : ι → Ordinal} (H : ∀ i, f i < iSup f) :
cof (iSup f) ≤ #ι := by
rw [← (#ι).lift_id]
exact cof_iSup_le_lift H
theorem iSup_lt_ord_lift {ι} {f : ι → Ordinal} {c : Ordinal} (hι : Cardinal.lift.{v, u} #ι < c.cof)
(hf : ∀ i, f i < c) : iSup f < c ... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 262 | 282 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... |
@[simp]
theorem pi_inter_compl (s : Set ι) : pi s t ∩ pi sᶜ t = pi univ t := by
rw [← union_pi, union_compl_self]
| Mathlib/Data/Set/Prod.lean | 753 | 757 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | · use 1
rw [pow_zero] at h ⊢
use Units.mkOfMulEqOne _ _ h
rw [Units.val_mkOfMulEqOne, one_mul]
have hc : c ∣ a * b := by
rw [h]
exact dvd_pow_self _ hk.ne'
obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
use d₁
obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc ... | Mathlib/Algebra/GCDMonoid/Basic.lean | 563 | 572 |
/-
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... | Mathlib/LinearAlgebra/Lagrange.lean | 628 | 631 | |
/-
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.Group.Basic
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Logic.Equi... | simp only [Function.comp_def, functor_norm, traverse_mul]
naturality := fun η α β f x ↦
recOnPure x (fun x ↦ by simp only [traverse_pure, functor_norm, Function.comp])
(fun x y ih1 ih2 ↦ by simp only [traverse_mul, functor_norm, ih1, ih2])
traverse_eq_map_id := fun f x ↦
FreeSemi... | Mathlib/Algebra/Free.lean | 668 | 677 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | rw [setOf_sOppSide_eq_image2 hx hp, ← Set.image_prod]
refine (isConnected_Iio.prod (isConnected_iff_connectedSpace.2 ?_)).image _
((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn
convert AddTorsor.connectedSpace s.direction s
theorem isPreconnected_setOf_sOppSide (s : AffineSubspace ... | Mathlib/Analysis/Convex/Side.lean | 803 | 819 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | simp [factorization_eq_zero_of_non_prime n hp]
@[deprecated (since := "2024-10-24")] alias ord_compl_of_not_prime := ordCompl_of_not_prime
| Mathlib/Data/Nat/Factorization/Basic.lean | 99 | 102 |
/-
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.CategoryTheory.Category.Pairwise
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
import Mathlib.CategoryTheory.Limits.Final
import Mathli... | · apply interUnionPullbackConeLift_left
· apply interUnionPullbackConeLift_right
· intro m h₁ h₂
rw [← cancel_mono (F.1.map (eqToHom hι.symm).op)]
apply (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 ι).some.hom_ext
rintro ((_ | _) | (_ | _)) <;>
rw [Category.assoc, Category.assoc]
| Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | 393 | 399 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
/-!
# Decompositions of additive monoids, groups, and modules into dire... | theorem degree_eq_of_mem_mem {x : M} {i j : ι} (hxi : x ∈ ℳ i) (hxj : x ∈ ℳ j) (hx : x ≠ 0) :
i = j := by
contrapose! hx; rw [← decompose_of_mem_same ℳ hxj, decompose_of_mem_ne ℳ hxi hx]
| Mathlib/Algebra/DirectSum/Decomposition.lean | 140 | 142 |
/-
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.Arctan
import Mathlib.Analysis.SpecialFunctions.... |
theorem hasDerivAt_tan {x : ℝ} (h : cos x ≠ 0) : HasDerivAt tan (1 / cos x ^ 2) x :=
| Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | 26 | 27 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | library_note "fact non-instances"/--
In most cases, we should not have global instances of `Fact`; typeclass search only reads the head
| Mathlib/Logic/Basic.lean | 85 | 86 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!... | · rw [prod_take_succ _ _ h]
exact le_self_mul
· simp [take_of_length_le h, take_of_length_le (le_trans h (Nat.le_succ _))]
/-- See also `List.single_le_prod`. -/
| Mathlib/Algebra/Order/BigOperators/Group/List.lean | 193 | 197 |
/-
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 | 1,128 | 1,134 | |
/-
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.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we have `image_le_ker... |
theorem imageToKernel_comp_left {Z : V} (h : Z ⟶ A) (w : f ≫ g = 0) :
imageToKernel (h ≫ f) g (by simp [w]) =
Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫ imageToKernel f g w := by
ext
| Mathlib/Algebra/Homology/ImageToKernel.lean | 101 | 105 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | Mathlib/Data/Seq/Computation.lean | 1,248 | 1,256 | |
/-
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.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | · rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
have : countablePartitionSet n x ×ˢ univ = {p : γ × β | p.1 ∈ countablePartitionSet n x} := by
ext x
simp
rw [this, ENNReal.div_self]
· simp
· rwa [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)] at h
· exact mea... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 651 | 672 |
/-
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
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Instances.ENNReal.Lemma... | Summable f ↔ (∀ x, Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' y, f ⟨x, y⟩ := by
lift f to (Σx, β x) → ℝ≥0 using hf
simpa using mod_cast NNReal.summable_sigma
| Mathlib/Topology/Algebra/InfiniteSum/Real.lean | 67 | 70 |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... |
theorem lmarginal_eq_of_subset {f g : (∀ i, X i) → ℝ≥0∞} (hst : s ⊆ t)
(hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ) :
∫⋯∫⁻_t, f ∂μ = ∫⋯∫⁻_t, g ∂μ := by
rw [← union_sdiff_of_subset hst, lmarginal_union' μ f hf disjoint_sdiff,
| Mathlib/MeasureTheory/Integral/Marginal.lean | 223 | 227 |
/-
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.Integral.DominatedConvergence
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# Measurability of the integral against a kernel
Th... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 204 | 207 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,481 | 1,481 | |
/-
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.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... |
variable [SemilatticeInf α] {a b c : α}
/-- An inf-irreducible element is a non-top element which isn't the infimum of anything bigger. -/
def InfIrred (a : α) : Prop :=
¬IsMax a ∧ ∀ ⦃b c⦄, b ⊓ c = a → b = a ∨ c = a
/-- An inf-prime element is a non-top element which isn't bigger than the infimum of anything
bigge... | Mathlib/Order/Irreducible.lean | 130 | 143 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
/-!
# Engel's theorem
This file contains a proof of Engel's theorem providing necessary and suf... | rw [← LieSubalgebra.toSubmodule_le_toSubmodule]
exact sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) hK₂.le
refine ⟨K', ?_, lt_iff_le_and_ne.mpr ⟨hKK', fun contra => hx₂ (contra.symm ▸ hxK')⟩⟩
intro M _i1 _i2 _i3 _i4 h
obtain ⟨I, hI₁ : (I : LieSubalgebra R K') = LieSubalgebra.ofLe hKK'⟩ :=
... | Mathlib/Algebra/Lie/Engel.lean | 192 | 216 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m)
| Mathlib/SetTheory/Ordinal/Notation.lean | 811 | 811 |
/-
Copyright (c) 2020 Ruben Van de Velde. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ruben Van de Velde
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
/-!
# Extending a ... | rw [← mul_le_mul_left (norm_pos_iff.2 h), ← sq]
calc
‖lm x‖ ^ 2 = fr (conj (lm x : 𝕜) • x) := fr.toLinearMap.norm_extendTo𝕜'_apply_sq x
_ ≤ ‖fr (conj (lm x : 𝕜) • x)‖ := le_abs_self _
_ ≤ ‖fr‖ * ‖conj (lm x : 𝕜) • x‖ := le_opNorm _ _
_ = ‖(lm x : 𝕜)‖ * (‖fr‖ * ‖x‖) := by rw [norm_smul, norm_con... | Mathlib/Analysis/NormedSpace/Extend.lean | 109 | 120 |
/-
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.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... | Multiset.lcm_singleton
@[local simp] -- This will later be provable by other `simp` lemmas.
theorem normalize_lcm : normalize (s.lcm f) = s.lcm f := by simp [lcm_def]
theorem lcm_union [DecidableEq β] : (s₁ ∪ s₂).lcm f = GCDMonoid.lcm (s₁.lcm f) (s₂.lcm f) :=
| Mathlib/Algebra/GCDMonoid/Finset.lean | 77 | 82 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... |
/-- If the second of three points is weakly between the other two, the oriented angle at the
first point is zero. -/
theorem _root_.Wbtw.oangle₂₁₃_eq_zero {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) : ∡ p₂ p₁ p₃ = 0 := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 420 | 423 |
/-
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... | exact (hε.not_le hs).elim
obtain rfl | rfl := Finset.subset_singleton_iff.1 ht'
· replace ht : ε ≤ 0 := by simpa using ht
exact (hε.not_le ht).elim
· rwa [sub_self, abs_zero]
theorem not_isUniform_zero : ¬G.IsUniform (0 : 𝕜) s t := fun h =>
(abs_nonneg _).not_lt <| h (empty_subset _) (empty_subset _) ... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 100 | 109 |
/-
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.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... | Mathlib/Data/NNRat/Defs.lean | 426 | 430 | |
/-
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.SetLike.Basic
import Mathlib.ModelTheory.Semantics
/-!
# Definable Sets
This file defines what it means for a set over a first-order structure t... | ext b
simp only [Function.comp_apply, Equiv.apply_symm_apply]
· intro i
ext a
simp
| Mathlib/ModelTheory/Definability.lean | 167 | 172 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... | Mathlib/Data/Complex/Module.lean | 545 | 548 | |
/-
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 | 670 | 673 | |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Order.CompleteLattice.Finset
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathl... |
@[to_additive]
theorem mulIndicator_eventuallyEq (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t) :
mulIndicator s f =ᶠ[l] mulIndicator t g :=
(eventually_inf_principal.1 hf).mp <| hs.mem_iff.mono fun x hst hfg =>
by_cases
| Mathlib/Order/Filter/IndicatorFunction.lean | 28 | 33 |
/-
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.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | instance : CoeOut X.Opens Scheme := ⟨toScheme⟩
| Mathlib/AlgebraicGeometry/Restrict.lean | 47 | 48 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Ring.Equiv
/-!
# Propositional typeclasses on several ring homs
This file contains three typeclasses used in the definitio... | Mathlib/Algebra/Ring/CompTypeclasses.lean | 188 | 192 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
import Mathlib.Algebra.Ring.Action.Rat
import Mathlib.Tactic.FastInstance
/-!
# Cauchy completion
This fi... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 414 | 437 | |
/-
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... | intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
obtain ⟨r, hr⟩ := this.mp hx
have := smul_left_injective _ hx' (hr.trans (one_smul _ x).symm)
exact I.eq_top_iff_one.not.mp h (this ▸ r.prop)
/-- **Krull's intersection theorem** for noetherian domains. -/
| Mathlib/RingTheory/Filtration.lean | 461 | 468 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | parts_pos {n hn} := by
rcases mem_add.mp hn with hn | hn
· exact zero_lt_one.trans (one_lt_of_mem_cycleType hn)
· exact lt_of_lt_of_le zero_lt_one (ge_of_eq (Multiset.eq_of_mem_replicate hn))
parts_sum := by
rw [sum_add, sum_cycleType, Multiset.sum_replicate, nsmul_eq_mul, Nat.cast_id, mul_one,
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 553 | 558 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | finprod_congr fun _ => finprod_true _
variable {f g : α → M} {a b : α} {s t : Set α}
@[to_additive]
theorem finprod_mem_congr (h₀ : s = t) (h₁ : ∀ x ∈ t, f x = g x) :
| Mathlib/Algebra/BigOperators/Finprod.lean | 495 | 500 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | · have hn : Nonempty n := Fintype.card_pos_iff.mp hc
refine nonsing_inv_apply_not_isUnit _ ?_
simp [hn]
noncomputable instance : InvOneClass (Matrix n n α) :=
{ Matrix.one, Matrix.inv with inv_one := inv_eq_left_inv (by simp) }
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 496 | 501 |
/-
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.Category.MonCat.ForgetCorepresentable
import Mathlib.Algebra.Category.Grp.ForgetCorepresentable
import Mathlib.Algebra.Category.Grp.Preadditive
im... | Mathlib/Algebra/Category/Grp/Limits.lean | 550 | 553 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | | s :: l₁, l₂, h₁ => by
if h₂ : a = s.1 then simp [h₂]
else simp at h₁; rcases h₁ with h₁ | h₁ <;>
[exact absurd h₁ h₂; simp [h₂, kerase_append_left h₁]]
theorem kerase_append_right {a} :
∀ {l₁ l₂ : List (Sigma β)}, a ∉ l₁.keys → kerase a (l₁ ++ l₂) = l₁ ++ kerase a l₂
| [], _, _ => rfl
| _ :: ... | Mathlib/Data/List/Sigma.lean | 491 | 500 |
/-
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.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
/-!
# Definition... |
variable [Semiring R]
| Mathlib/RingTheory/Nilpotent/Defs.lean | 235 | 237 |
/-
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.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | theorem mem_powers_of_prime_card {p : ℕ} [hp : Fact p.Prime]
(h : Nat.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ Submonoid.powers g := by
have : Finite G := Nat.finite_of_card_ne_zero (h ▸ hp.1.ne_zero)
rw [mem_powers_iff_mem_zpowers]
exact mem_zpowers_of_prime_card h hg
@[to_additive]
theorem powers_eq_top_... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 179 | 186 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Jacobson.Ideal
import Mathlib.RingTheory.Ideal.Quo... |
@[simp]
theorem add_apply (n : ℕ) (f g : AdicCauchySequence I M) : (f + g) n = f n + g n :=
rfl
@[simp]
theorem sub_apply (n : ℕ) (f g : AdicCauchySequence I M) : (f - g) n = f n - g n :=
rfl
@[simp]
theorem smul_apply (n : ℕ) (r : R) (f : AdicCauchySequence I M) : (r • f) n = r • f n :=
rfl
| Mathlib/RingTheory/AdicCompletion/Basic.lean | 407 | 418 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
/-!
# Some results on tensor product of s... | mulMap M N ∘ₗ (inclusion hN).lTensor M = mulMap M N' :=
TensorProduct.ext' fun _ _ ↦ rfl
variable {M N} in
theorem mulMap_comp_map_inclusion {M' N' : Submodule R S} (hM : M' ≤ M) (hN : N' ≤ N) :
mulMap M N ∘ₗ TensorProduct.map (inclusion hM) (inclusion hN) = mulMap M' N' :=
TensorProduct.ext' fun _ _ ↦ rfl... | Mathlib/LinearAlgebra/TensorProduct/Submodule.lean | 98 | 104 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | subst hf
exact IsZero.of_mono_zero X Y
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 205 | 207 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,220 | 1,224 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Construction
/-!
# Predicate for localized categories
In this file, a predicate `L.IsLocalization W` is introduced for a functor `... | variable {W f g}
lemma map_eq (h : AreEqualizedByLocalization W f g) (L : C ⥤ D) [L.IsLocalization W] :
L.map f = L.map g :=
(areEqualizedByLocalization_iff L W f g).1 h
end AreEqualizedByLocalization
end
end CategoryTheory
| Mathlib/CategoryTheory/Localization/Predicate.lean | 494 | 507 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | @[reassoc (attr := simp)]
lemma RightHomologyData.homologyIso_hom_comp_rightHomologyIso_inv (h : S.RightHomologyData) :
h.homologyIso.hom ≫ h.rightHomologyIso.inv = S.rightHomologyIso.inv := by
dsimp only [homologyIso]
simp only [Iso.trans_hom, Iso.symm_hom, assoc, Iso.hom_inv_id, comp_id]
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 976 | 981 |
/-
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 | 1,730 | 1,740 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | Mathlib/Analysis/Normed/Group/Basic.lean | 508 | 508 | |
/-
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... |
/-- **Minkowski inequality**: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expres... | Mathlib/Analysis/MeanInequalities.lean | 857 | 886 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 411 | 413 | |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... |
theorem tan_pi_div_two_sub (x : ℝ) : tan (π / 2 - x) = (tan x)⁻¹ := by
rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos, inv_div, sin_pi_div_two_sub, cos_pi_div_two_sub]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 921 | 924 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | ext u
simp only [mem_infs, mem_powerset, le_eq_subset, inf_eq_inter]
refine ⟨fun h ↦ ⟨_, inter_subset_left (s₂ := u), _, inter_subset_left (s₂ := u), ?_⟩, ?_⟩
| Mathlib/Data/Finset/Sups.lean | 368 | 370 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | apply nilRankAux_basis_indep
lemma polyCharpoly_coeff_nilRank_ne_zero :
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 471 | 473 |
/-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.Flat.Basic
import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
import Mathlib.RingTheory.QuotSMulTop... |
/-- Given a left exact sequence `0 → M → M' → M''`, if `r` is regular on both
`M` and `M''` it's regular `M'` too. -/
lemma isSMulRegular_of_range_eq_ker {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''}
(hf : Function.Injective f) (hfg : LinearMap.range f = LinearMap.ker g)
(h1 : IsSMulRegular M r) (h2 : IsSMulRegular M'' ... | Mathlib/RingTheory/Regular/IsSMulRegular.lean | 112 | 123 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 548 | 550 | |
/-
Copyright (c) 2023 Kyle Miller, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Rémi Bottinelli
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Set.Card
/-!
# Connectivity of subgraphs and induced graphs
## Main de... | @[simp]
theorem verts_toSubgraph (p : G.Walk u v) : p.toSubgraph.verts = { w | w ∈ p.support } :=
Set.ext fun _ => p.mem_verts_toSubgraph
| Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 158 | 160 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | (hP1 : IsPGroup p P) (hP2 : ¬ p ∣ P.index) : Sylow p G :=
| Mathlib/GroupTheory/Sylow.lean | 76 | 76 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
We gather results abou... | (h : Irreducible <| cyclotomic n K) [NeZero (n : K)] : cyclotomic n K = minpoly K μ := by
haveI := NeZero.of_faithfulSMul K R n
refine minpoly.eq_of_irreducible_of_monic h ?_ (cyclotomic.monic n K)
rwa [aeval_def, eval₂_eq_eval_map, map_cyclotomic, ← IsRoot.def, isRoot_cyclotomic_iff]
| Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 165 | 168 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | Prod.ext_iff] at hp
obtain ⟨q₁, q₅, q₂⟩ := p
dsimp only at hp ⊢
obtain ⟨⟨⟨q, hq⟩, v⟩, hv, rfl, rfl, rfl⟩ := hp
dsimp only [Subtype.coe_mk, Set.mem_setOf] at hv ⊢
obtain ⟨hvr, hv0⟩ := hv
rw [← exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp₁p₂.symm)] at hvr
obtain ⟨r, ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 637 | 645 |
/-
Copyright (c) 2022 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.Group.Measure
/-!
# Lebesgue Integration on Groups
We develop properties of integrals with a group as domain.
This file contains pr... | theorem lintegral_inv_eq_self [IsInvInvariant μ] (f : G → ℝ≥0∞) :
∫⁻ x, f x⁻¹ ∂μ = ∫⁻ x, f x ∂μ := by
simpa using (lintegral_map_equiv f (μ := μ) <| MeasurableEquiv.inv G).symm
| Mathlib/MeasureTheory/Group/LIntegral.lean | 34 | 37 |
/-
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... | (Orientation.map (Fin 2) φ.toLinearEquiv o).rightAngleRotation =
(φ.symm.trans o.rightAngleRotation).trans φ :=
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 317 | 318 |
/-
Copyright (c) 2021 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.SetFamily.Shadow
/-!
# UV-compressions
This file defines UV-compression. It is an operation on a set family that reduces its ... | rintro a ha b hb hab
have h : ((a ⊔ u) \ v) \ u ⊔ v = ((b ⊔ u) \ v) \ u ⊔ v := by
dsimp at hab
rw [hab]
rwa [sdiff_sdiff_comm, ha.1.symm.sup_sdiff_cancel_right, sdiff_sdiff_comm,
hb.1.symm.sup_sdiff_cancel_right, sdiff_sup_cancel ha.2, sdiff_sup_cancel hb.2] at h
-- The namespace is here to distingui... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 57 | 64 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 914 | 920 | |
/-
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... | by_cases hq : q.Monic
· rcases subsingleton_or_nontrivial R with hR | hR
· simp only [eq_iff_true_of_subsingleton]
· exact
(div_modByMonic_unique (c • (p /ₘ q)) (c • (p %ₘ q)) hq
⟨by rw [mul_smul_comm, ← smul_add, modByMonic_add_div p hq],
| Mathlib/Algebra/Polynomial/RingDivision.lean | 50 | 55 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | third point is zero. -/
theorem _root_.Sbtw.oangle₂₃₁_eq_zero {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) : ∡ p₂ p₃ p₁ = 0 :=
h.wbtw.oangle₂₃₁_eq_zero
/-- If the second of three points is weakly between the other two, the oriented angle at the
third point (reversed) is zero. -/
theorem _root_.Wbtw.oangle₁₃₂_eq_zero {p₁ p₂ ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 450 | 456 |
/-
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... | @[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ᵐ i = (f i)⁺ᵐ := rfl
@[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ᵐ i = (f i)⁻ᵐ := rfl
| Mathlib/Algebra/Order/Group/PosPart.lean | 247 | 249 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... |
section deriv
variable {f g : ℝ → ℝ} {f' g' x y p : ℝ} {s : Set ℝ}
| Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 585 | 589 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Chris Hughes, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Int.Cast.Prod
import Mathlib.Algebra.GroupWithZero.Prod
import Mathlib.Algebra.Ring.CompTypeclasse... | Mathlib/Algebra/Ring/Prod.lean | 379 | 386 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Subobject.FactorThru
import Mathlib.CategoryTheory.Subobject.WellPowered
import... | section OrderBot
| Mathlib/CategoryTheory/Subobject/Lattice.lean | 264 | 265 |
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