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) 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... | Mathlib/Data/Set/Function.lean | 1,585 | 1,592 | |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... |
theorem image_ennreal_ofReal (h : s.OrdConnected) : (ENNReal.ofReal '' s).OrdConnected := by
| Mathlib/Data/ENNReal/Basic.lean | 718 | 719 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... | exact epi_of_epi_fac this
· intro
rw [← this]
apply epi_comp
theorem exact_iff_epi_imageToKernel : S.Exact ↔ Epi (imageToKernel S.f S.g S.zero) := by
rw [S.exact_iff_epi_imageToKernel']
| Mathlib/CategoryTheory/Abelian/Exact.lean | 57 | 63 |
/-
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.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | have h : ⟪x, (A† ∘L A) x⟫ = ⟪A x, A x⟫ := by rw [← adjoint_inner_right]; rfl
rw [h, ← inner_self_eq_norm_sq (𝕜 := 𝕜) _]
theorem apply_norm_eq_sqrt_inner_adjoint_right (A : E →L[𝕜] F) (x : E) :
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 144 | 147 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,689 | 1,693 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... | /-- If `S / R` and `S' / R'` are algebras, `i : R' →+* R` and `j : S →+* S'` are injective ring
homomorphisms, such that `R' → R → S → S'` and `R' → S'` commute, then the rank of `S / R` is
smaller than or equal to the rank of `S' / R'`. -/
theorem lift_rank_le_of_injective_injective
(i : R' →+* R) (j : S →+* S') (... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 210 | 216 |
/-
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... | rw [bodd_bit, div2_bit]
simp
@[simp]
lemma bitwise_bit {f : Bool → Bool → Bool} (h : f false false = false := by rfl) (a m b n) :
bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) := by
conv_lhs => unfold bitwise
| Mathlib/Data/Nat/Bitwise.lean | 75 | 81 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... |
@[simp]
theorem mem_inf (x : M) : x ∈ N ⊓ N' ↔ x ∈ N ∧ x ∈ N' := by
rw [← mem_toSubmodule, ← mem_toSubmodule, ← mem_toSubmodule, inf_toSubmodule,
| Mathlib/Algebra/Lie/Submodule.lean | 547 | 550 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | lift_ι_apply _ _ m
variable (Q₁) in
| Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 299 | 301 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... | (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
refine integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ ?_ ?_
· rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condExpL2 E' 𝕜 hm f)) f).symm)]
exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 282 | 300 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 715 | 717 | |
/-
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.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | lemma congr_φK {γ₁ γ₂ : LeftHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) : γ₁.φK = γ₂.φK := by rw [eq]
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 315 | 315 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | theorem map_one_fst_of_isMulTwoCocycle {f : G × G → M} (hf : IsMulTwoCocycle f) (g : G) :
f (1, g) = f (1, 1) := by
simpa only [one_smul, one_mul, mul_one, mul_right_inj] using (hf 1 1 g).symm
theorem map_one_snd_of_isMulTwoCocycle {f : G × G → M} (hf : IsMulTwoCocycle f) (g : G) :
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 594 | 598 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | Mathlib/NumberTheory/Pell.lean | 710 | 733 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 801 | 804 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... |
@[simp]
theorem det_smul_of_tower {α} [Monoid α] [MulAction α R] [IsScalarTower α R R]
[SMulCommClass α R R] (c : α) (A : Matrix n n R) :
det (c • A) = c ^ Fintype.card n • det A := by
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 267 | 271 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Group.Torsion
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
/-!
# Lemmas about the ... | lemma zpow_right_strictMono (ha : 1 < a) : StrictMono fun n : ℤ ↦ a ^ n := by
refine strictMono_int_of_lt_succ fun n ↦ ?_
rw [zpow_add_one]
exact lt_mul_of_one_lt_right' (a ^ n) ha
@[to_additive zsmul_pos] lemma one_lt_zpow (ha : 1 < a) (hn : 0 < n) : 1 < a ^ n := by
| Mathlib/Algebra/Order/Group/Basic.lean | 25 | 30 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Results about `minpoly... | section IsIntegral
variable (hx : IsIntegral R x)
include hx
lemma minpolyDiv_ne_zero [Nontrivial S] : minpolyDiv R x ≠ 0 := by
intro e
| Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean | 98 | 103 |
/-
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.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | obtain ⟨f, rfl⟩ := Hom.mk_surjective f
rw [show Hom.mk f = homMk (mk f.f f.s f.hs) by rfl,
← Q_map_comp_Qinv f.f f.s f.hs, F₁.map_comp, F₂.map_comp, assoc]
erw [Functor.congr_hom h f.f]
rw [assoc, assoc]
congr 2
have := inverts W _ f.hs
rw [← cancel_epi (F₂.map ((Q W).map f.s)), ← F₂.m... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 608 | 625 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... |
theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ)
(hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) :
⨍⁻ x, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) := by
simpa only [union_compl_self, restrict_univ] using
laverage_union_mem_openSegment aedisjoint_comp... | Mathlib/MeasureTheory/Integral/Average.lean | 205 | 215 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.IdentDistrib
import Mathlib.Probability.Independence.Integrable
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mat... | ext i
exact (hident i).truncation.integral_eq
convert Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 A) using 1
ext1 n
simp only [sum_sub_distrib, sum_const, card_range, nsmul_eq_mul, sum_apply, sub_left_inj]
rw [integral_finset_sum _ fun i _ => ?_]
exact ((hident i).symm... | Mathlib/Probability/StrongLaw.lean | 505 | 517 |
/-
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.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... |
end NeZero
/-!
### Exponential characteristic
| Mathlib/Algebra/CharP/Defs.lean | 292 | 296 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Modules over a ring
In th... | Mathlib/Algebra/Module/Defs.lean | 241 | 245 | |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | rfl
/-- See also `LieModule.traceForm_eq_sum_finrank_nsmul'` for an expression omitting the zero
weights. -/
lemma traceForm_eq_sum_finrank_nsmul :
traceForm K L M = ∑ χ : Weight K L M, finrank K (genWeightSpace M χ) •
(χ : L →ₗ[K] K).smulRight (χ : L →ₗ[K] K) := by
ext
| Mathlib/Algebra/Lie/TraceForm.lean | 415 | 422 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Order.ScottContinuity
import Mathlib.Topology.Order.UpperLowerSetTopology
/-!
# Scott topology
This file introduces the Scott topology on a p... | lemma DirSupInacc.union (hs : DirSupInacc s) (ht : DirSupInacc t) : DirSupInacc (s ∪ t) := by
rw [← dirSupClosed_compl, compl_union]; exact hs.compl.inter ht.compl
| Mathlib/Topology/Order/ScottTopology.lean | 111 | 113 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | theorem det_diagonal {d : n → R} : det (diagonal d) = ∏ i, d i := by
rw [det_apply']
refine (Finset.sum_eq_single 1 ?_ ?_).trans ?_
· rintro σ - h2
obtain ⟨x, h3⟩ := not_forall.1 (mt Equiv.ext h2)
convert mul_zero (ε σ)
apply Finset.prod_eq_zero (mem_univ x)
exact if_neg h3
· simp
· simp
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 73 | 82 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... | (hm : ∀ x y, f (midpoint R x y) = midpoint R' (f x) (f y)) : E →+ F where
toFun := f
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 226 | 227 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
/-!
# One-dimensional de... |
nonrec theorem HasDerivWithinAt.neg (h : HasDerivWithinAt f f' s x) :
| Mathlib/Analysis/Calculus/Deriv/Add.lean | 211 | 212 |
/-
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]
theorem affineSegment_image (f : P →ᵃ[R] P') (x y : P) :
| Mathlib/Analysis/Convex/Between.lean | 45 | 46 |
/-
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.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 681 | 684 | |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic... | rw [← str_hom_commute, str_eq_of_le_nhds _ x _]
apply h
| Mathlib/Topology/Category/Compactum.lean | 363 | 365 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... |
@[simp]
theorem dropFun_comp {α₀ α₁ α₂ : TypeVec (n + 1)} (f₀ : α₀ ⟹ α₁) (f₁ : α₁ ⟹ α₂) :
| Mathlib/Data/TypeVec.lean | 229 | 231 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | ⟨by contrapose!; exact Gamma_ne_zero, by rintro ⟨m, rfl⟩; exact Gamma_neg_nat_eq_zero m⟩
| Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 531 | 532 |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.Tactic.ApplyFun
/-!
# Rigid (autonomous) monoida... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 622 | 624 | |
/-
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... | calc
x ^ n / n ! ≤ ∑ k ∈ range (n + 1), x ^ k / k ! :=
single_le_sum (f := fun k ↦ x ^ k / k !) (fun k _ ↦ by positivity) (self_mem_range_succ n)
_ ≤ exp x := sum_le_exp_of_nonneg hx _
| Mathlib/Data/Complex/Exponential.lean | 247 | 250 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,623 | 1,625 | |
/-
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.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Data.Nat.Basic
/-!
# Partial predecessor and partial subtraction on the natur... | | 0 => by simp
| n + 1 => by constructor <;> intro <;> contradiction
| Mathlib/Data/Nat/PSub.lean | 63 | 65 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
/-!
# Traversing collections
This file proves basic properties of traversable and applicative ... | (Comp.mk ∘ map (traverse h) ∘ traverse g : t α → Comp F G (t γ)) := by
ext
| Mathlib/Control/Traversable/Lemmas.lean | 98 | 99 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | theorem cantorFunctionAux_eq (h : f n = g n) :
cantorFunctionAux c f n = cantorFunctionAux c g n := by simp [cantorFunctionAux, h]
| Mathlib/Data/Real/Cardinality.lean | 73 | 75 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... |
/-- `SignType.cast` as a `MulWithZeroHom`. -/
@[simps]
def castHom {α} [MulZeroOneClass α] [HasDistribNeg α] : SignType →*₀ α where
toFun := cast
| Mathlib/Data/Sign.lean | 258 | 262 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... |
theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s :=
starConvex_zero_iff.2 fun _ hx a ha₀ ha₁ =>
hs _ (by rwa [Real.norm_of_nonneg ha₀]) (smul_mem_smul_set hx)
theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
a ≤ gauge s x := by
rw [starConve... | Mathlib/Analysis/Convex/Gauge.lean | 201 | 212 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... |
alias ⟨_, Odd.pow_nonpos⟩ := Odd.pow_nonpos_iff
| Mathlib/Algebra/Order/Ring/Basic.lean | 261 | 262 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Graded tensor products over graded algebras
The graded tens... |
theorem auxEquiv_symm_one : (auxEquiv R 𝒜 ℬ).symm 1 = 1 :=
(LinearEquiv.symm_apply_eq _).mpr (auxEquiv_one _ _).symm
| Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 133 | 135 |
/-
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.HomologicalComplexLimits
import Mathlib.Algebra.Homology.Additive
/-! Binary biproducts of homological complexes
In this file, it is shown tha... | @[reassoc (attr := simp)]
lemma biprod_inl_fst_f (i : ι) :
(biprod.inl : K ⟶ K ⊞ L).f i ≫ (biprod.fst : K ⊞ L ⟶ K).f i = 𝟙 _ := by
rw [← comp_f, biprod.inl_fst, id_f]
| Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean | 65 | 68 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.LinearAlgebra.Finsupp.SumPr... | decomposeAux f ↑x = DirectSum.of (fun i => gradeBy R f i) i x := by
classical
obtain ⟨x, hx⟩ := x
revert hx
refine Finsupp.induction x ?_ ?_
· intro hx
symm
exact AddMonoidHom.map_zero _
· intro m b y hmy hb ih hmby
have : Disjoint (Finsupp.single m b).support y.support := by
simpa onl... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 140 | 150 |
/-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Catego... | variable (C)
| Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 323 | 324 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 2,212 | 2,224 | |
/-
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.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | induction' n with n IH generalizing xs
· simpa using formPerm_apply_getElem_zero _ h _
· rcases xs with (_ | ⟨x, _ | ⟨y, l⟩⟩)
| Mathlib/GroupTheory/Perm/List.lean | 160 | 162 |
/-
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.Set.Image
import Mathlib.Topology.Bases
import Mathlib.Topology.Inseparable
import Mathlib.Topology.Compactness.Exterior
/-!
# Alexandrov-discrete to... | alexandrovDiscrete_coinduced
| Mathlib/Topology/AlexandrovDiscrete.lean | 194 | 195 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | Mathlib/Data/Part.lean | 774 | 776 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... | obtain rfl | hb' := hb.eq_or_lt
· rw [add_zero] at hab
rwa [hab, zero_smul, one_smul, add_zero]
obtain rfl | hxy := eq_or_ne x y
· rwa [Convex.combo_self hab]
exact h hy hxy ha' hb' hab
theorem starConvex_iff_openSegment_subset [ZeroLEOneClass 𝕜] (hx : x ∈ s) :
StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → ope... | Mathlib/Analysis/Convex/Star.lean | 178 | 191 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.FieldTheory.Perfect
import Mathlib.LinearAlgebra.AnnihilatingPolynomial
import Mathlib.RingTheory.Artinian.Instances
im... | Mathlib/LinearAlgebra/Semisimple.lean | 77 | 77 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | theorem coprime_sub_self_right {m n : ℕ} (h : m ≤ n) : Coprime m (n - m) ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_sub_self_right h]
| Mathlib/Data/Nat/GCD/Basic.lean | 177 | 178 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Yaël Dillies
-/
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Group
import Mathlib.Tac... | /-- Any action on `ℝ` which factors through `ℝ≥0` applies to an `AddGroupSeminorm`. -/
instance toSMul : SMul R (AddGroupSeminorm E) :=
⟨fun r p =>
{ toFun := fun x => r • p x
map_zero' := by
simp only [← smul_one_smul ℝ≥0 r (_ : ℝ), NNReal.smul_def, smul_eq_mul, map_zero, mul_zero]
| Mathlib/Analysis/Normed/Group/Seminorm.lean | 396 | 401 |
/-
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.SetTheory.Game.State
/-!
# Domineering as a combinatorial game.
We define the game of Domineering, played on a chessboard of arbitrary shape
(possibly ev... | Finset.card (moveLeft b m) / 2 < Finset.card b / 2 := by simp [← moveLeft_card h, lt_add_one]
theorem moveRight_smaller {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) :
Finset.card (moveRight b m) / 2 < Finset.card b / 2 := by simp [← moveRight_card h, lt_add_one]
/-- The instance describing allowed moves on a Dom... | Mathlib/SetTheory/Game/Domineering.lean | 109 | 114 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Robin Carlier
-/
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
/... | theorem hasCoequalizer_of_common_section [HasReflexiveCoequalizers C] {A B : C} {f g : A ⟶ B}
(r : B ⟶ A) (rf : r ≫ f = 𝟙 _) (rg : r ≫ g = 𝟙 _) : HasCoequalizer f g := by
letI := IsReflexivePair.mk' r rf rg
infer_instance
| Mathlib/CategoryTheory/Limits/Shapes/Reflexive.lean | 154 | 157 |
/-
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.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 465 | 477 | |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Additive.Corner.Defs
import Mathlib.Combinatorics.SimpleGraph... | The maximum density of a corner-free set in `G × G` goes to zero as `|G|` tends to infinity. -/
theorem corners_theorem (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε ≤ card G)
(A : Finset (G × G)) (hAε : ε * card G ^ 2 ≤ #A) : ¬ IsCornerFree (A : Set (G × G)) := by
rintro hA
rw [cornersTheoremBound, Nat.add_... | Mathlib/Combinatorics/Additive/Corner/Roth.lean | 79 | 96 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.SplitIfs
import Mathlib.Logic.Basic
/-!
# Typeclasses for groups with an... | Mathlib/Algebra/GroupWithZero/Defs.lean | 290 | 290 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,995 | 1,996 | |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.... | IsNat a a' → IsNat b b' → Nat.div a' b' = c → IsNat (a / b) c
| _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, rfl => ⟨by aesop⟩
| Mathlib/Tactic/NormNum/Basic.lean | 553 | 555 |
/-
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.Algebra.Group.Submonoid.Pointwise
/-!
# Submonoid of inverses
Given a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid ... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 215 | 217 | |
/-
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 | 776 | 776 | |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... | sinZeta x (2 * k + 1) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by
rw [← (hasSum_nat_sinZeta x (?_ : 1 < re (2 * k + 1))).tsum_eq]
· refine Eq.trans ?_ <|
(congr_arg ofReal (hasSum_one_div_nat_pow_mul_sin h... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 76 | 97 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.OrderedM... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 615 | 617 | |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Order.Filter.Extr
/-!
# Lemmas about the `sup` and `in... | classical
refine s.induction ?_ ?_
· rw [Finset.prod_empty, Finset.sum_empty, one_def, supDegree_single]
split_ifs; exacts [bot_le, hzero.le]
· intro i s his ih
rw [Finset.prod_insert his, Finset.sum_insert his]
exact (supDegree_mul_le hadd).trans (by gcongr)
theorem apply_add_of_supDegree_le (hadd... | Mathlib/Algebra/MonoidAlgebra/Degree.lean | 296 | 312 |
/-
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 ... | rfl
@[simp]
theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' :=
rfl
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 405 | 410 |
/-
Copyright (c) 2023 Shogo Saito. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shogo Saito. Adapted for mathlib by Hunter Monroe
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.GCD.BigOperators
/-!
# Chinese Re... |
theorem chineseRemainderOfList_perm {l l' : List ι} (hl : l.Perm l')
(hs : ∀ i ∈ l, s i ≠ 0) (co : l.Pairwise (Coprime on s)) :
(chineseRemainderOfList a s l co : ℕ) =
chineseRemainderOfList a s l' (co.perm hl coprime_comm.mpr) := by
let z := chineseRemainderOfList a s l' (co.perm hl coprime_comm.mpr)
... | Mathlib/Data/Nat/ChineseRemainder.lean | 107 | 118 |
/-
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.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | def gluePremetric (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : PseudoMetricSpace (X ⊕ Y) where
dist := glueDist Φ Ψ 0
dist_self := glueDist_self Φ Ψ 0
dist_comm := glueDist_comm Φ Ψ 0
dist_triangle := glueDist_triangle Φ Ψ 0 fun p q => by rw [hΦ.dist_eq, hΨ.dist_eq]; simp
/-- Given two isometric embeddings `Φ : Z → X... | Mathlib/Topology/MetricSpace/Gluing.lean | 453 | 463 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int.Parity
/-!
# Results about powers in fields or... | Mathlib/Algebra/Field/Power.lean | 33 | 33 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | /-- Given an eventually-constant monotone sequence `a₀ ≤ a₁ ≤ a₂ ≤ ...` in a partially-ordered
type, `monotonicSequenceLimitIndex a` is the least natural number `n` for which `aₙ` reaches the
constant value. For sequences that are not eventually constant, `monotonicSequenceLimitIndex a`
is defined, but is a junk value.... | Mathlib/Order/OrderIsoNat.lean | 250 | 258 |
/-
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... | rw [← log_abs]
obtain hx | hx := (abs_nonneg x).eq_or_gt
· simp [hx]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 169 | 171 |
/-
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... |
theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 193 | 194 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 676 | 678 | |
/-
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.... | @[simp] lemma mk_le_swap : (b, a) ≤ x.swap ↔ (a, b) ≤ x := and_comm
end LE
| Mathlib/Order/Basic.lean | 1,071 | 1,073 |
/-
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.Grp.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
/-!
# The category of commutative additive groups has images.
Note that w... |
/-- the factorisation of any morphism in `AddCommGrp` through a mono. -/
def monoFactorisation : MonoFactorisation f where
I := image f
m := image.ι f
| Mathlib/Algebra/Category/Grp/Images.lean | 87 | 91 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.IndexNormal
import Mathlib.GroupTheory.Perm.F... | have h :
(⟨finRotate 5, finRotate_bit1_mem_alternatingGroup (n := 2)⟩ : alternatingGroup (Fin 5)) ∈
normalClosure _ :=
SetLike.mem_coe.1 (subset_normalClosure (Set.mem_singleton _))
-- Porting note: added `:` to help the elaborator
exact (mul_mem (Subgroup.normalClosure_norma... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 257 | 274 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 774 | 776 | |
/-
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.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... |
section Kernel
variable [HasZeroMorphisms C] (f : X ⟶ Y) [HasKernel f]
/-- The kernel of a morphism `f : X ⟶ Y` as a `Subobject X`. -/
| Mathlib/CategoryTheory/Subobject/Limits.lean | 72 | 77 |
/-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Circulant
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Fi... | theorem Coloring.even_length_iff_congr {α} {G : SimpleGraph α}
(c : G.Coloring Bool) {u v : α} (p : G.Walk u v) :
Even p.length ↔ (c u ↔ c v) := by
induction p with
| nil => simp
| Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean | 61 | 65 |
/-
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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | ((f.map m).roots.toFinset.toSet : Set S)).Finite :=
Set.Finite.biUnion
| Mathlib/Algebra/Polynomial/Roots.lean | 526 | 527 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | simp_rw [hasDerivAt_iff_hasFDerivAt] at hf ⊢
-- Now we need to rewrite hf' in terms of `ContinuousLinearMap`s. The tricky part is that
-- operator norms are written in terms of `≤` whereas metrics are written in terms of `<`. So we
-- need to shrink `ε` utilizing the archimedean property of `ℝ`
have hf' : Ten... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 496 | 522 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... |
This is a specialization of `EquicontinuousOn.tendsto_uniformOnFun_iff_pi'` to the case
where `𝔖` covers `X`. -/
theorem EquicontinuousOn.tendsto_uniformOnFun_iff_pi
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = univ)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f... | Mathlib/Topology/UniformSpace/Ascoli.lean | 354 | 364 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... |
end RCLike
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 281 | 282 |
/-
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.CategoryTheory.Comma.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | theorem diagonalObjPullbackFstIso_inv_snd_fst {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(diagonalObjPullbackFstIso f g).inv ≫ pullback.snd _ _ ≫ pullback.fst _ _ =
pullback.snd _ _ := by
delta diagonalObjPullbackFstIso
simp
| Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 345 | 349 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | alias _root_.CovBy.succ_eq := succ_eq_of_covBy
| Mathlib/Order/SuccPred/Basic.lean | 343 | 344 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | @[simp]
lemma exact_unop_iff (S : ShortComplex Cᵒᵖ) : S.unop.Exact ↔ S.Exact :=
S.unop.exact_op_iff.symm
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 191 | 193 |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
/-!
# ω-limits
For a function `ϕ : τ → α → β` where `β` is a topological space, we
define the ω-limit under `ϕ` of... | calc
_ ⊆ closure (image2 ϕ v s) := closure_mono (image2_subset inter_subset_right Subset.rfl)
_ ⊆ c := hv₂
· exact fun _ ↦ isClosed_closure
| Mathlib/Dynamics/OmegaLimit.lean | 288 | 292 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | theorem kernelIsoOfEq_trans {f g h : X ⟶ Y} [HasKernel f] [HasKernel g] [HasKernel h] (w₁ : f = g)
(w₂ : g = h) : kernelIsoOfEq w₁ ≪≫ kernelIsoOfEq w₂ = kernelIsoOfEq (w₁.trans w₂) := by
cases w₁; cases w₂; ext; simp [kernelIsoOfEq]
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 394 | 396 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | LineDifferentiableWithinAt 𝕜 f₀ s x v ↔ LineDifferentiableWithinAt 𝕜 f₁ s x v :=
⟨fun h' ↦ ((h.hasLineDerivWithinAt_iff hx).1 h'.hasLineDerivWithinAt).lineDifferentiableWithinAt,
fun h' ↦ ((h.hasLineDerivWithinAt_iff hx).2 h'.hasLineDerivWithinAt).lineDifferentiableWithinAt⟩
theorem Filter.EventuallyEq.lineD... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 343 | 347 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
The first part of the file pr... |
@[simp]
theorem linearEquivFunOnFintype_symm_single [Fintype ι] (i : ι) (m : M i) :
(linearEquivFunOnFintype R ι M).symm (Pi.single i m) = lof R ι M i m := by
change (DFinsupp.equivFunOnFintype.symm (Pi.single i m)) = _
| Mathlib/Algebra/DirectSum/Module.lean | 164 | 168 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | have f : RelEmbedding (Subrel r (· ∈ s)) fun a b : α => r a b ∧ a ∈ s ∧ b ∈ s :=
⟨⟨(↑), Subtype.coe_injective⟩, by simp⟩
refine ⟨fun h => ?_, f.wellFounded⟩
rw [WellFounded.wellFounded_iff_has_min]
intro t ht
by_cases hst : (s ∩ t).Nonempty
· rw [← Subtype.preimage_coe_nonempty] at hst
rcases h.has_... | Mathlib/Order/WellFoundedSet.lean | 77 | 89 |
/-
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... | Mathlib/Data/Set/Function.lean | 1,507 | 1,513 | |
/-
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.Data.Finset.Pi
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Set.Finite.Basic
/-!
# Preimage of ... | theorem subset_map_iff {f : α ↪ β} {s : Finset β} {t : Finset α} :
s ⊆ t.map f ↔ ∃ u ⊆ t, s = u.map f := by
classical
simp_rw [map_eq_image, subset_image_iff, eq_comm]
| Mathlib/Data/Finset/Preimage.lean | 113 | 116 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | rintro rfl; exact one_pos.not_le ha₀
theorem orderOf_abs_ne_one (h : |x| ≠ 1) : orderOf x = 0 := by
rw [orderOf_eq_zero_iff']
| Mathlib/GroupTheory/OrderOfElement.lean | 1,065 | 1,068 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Huỳnh Trần Khanh, Stuart Presnell
-/
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Math... | Mathlib/Data/Sym/Card.lean | 182 | 189 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | theorem HasDerivWithinAt.div_const (hc : HasDerivWithinAt c c' s x) (d : 𝕜') :
HasDerivWithinAt (fun x => c x / d) (c' / d) s x := by
simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 389 | 391 |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,464 | 1,467 | |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Algebra.Category.Grp.FilteredColimits
import Mathlib.Algebra.Category.ModuleCat.Colimits
/-!
# The forgetful functor from `R`-modules preserves filt... | { colimitMulAction F with
smul_zero := fun r => by
erw [colimit_zero_eq _ (IsFiltered.nonempty.some : J), colimit_smul_mk_eq, smul_zero]
rfl
zero_smul := fun x => by
| Mathlib/Algebra/Category/ModuleCat/FilteredColimits.lean | 100 | 104 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... |
lemma rpow_mul_natCast (x : ℝ≥0∞) (y : ℝ) (n : ℕ) : x ^ (y * n) = (x ^ y) ^ n := by
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 917 | 918 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.