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) 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.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | instance [SecondCountableTopology α] [SecondCountableTopology β] :
SecondCountableTopology (α ⊕ β) := by
let b :=
| Mathlib/Topology/Bases.lean | 898 | 900 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... |
variable (μ) in
lemma rnDeriv_tilted_left {ν : Measure α} [SigmaFinite μ] [SigmaFinite ν] (hfν : AEMeasurable f ν) :
(μ.tilted f).rnDeriv ν
=ᵐ[ν] fun x ↦ ENNReal.ofReal (exp (f x) / (∫ x, exp (f x) ∂μ)) * μ.rnDeriv ν x := by
let g := fun x ↦ ENNReal.ofReal (exp (f x) / (∫ x, exp (f x) ∂μ))
refine Measure... | Mathlib/MeasureTheory/Measure/Tilted.lean | 340 | 347 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... |
lemma rpow_sub_one' (hx : 0 ≤ x) (h : y - 1 ≠ 0) : x ^ (y - 1) = x ^ y / x := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 442 | 443 |
/-
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 | 2,160 | 2,162 | |
/-
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... | | 0, o => o
| oadd e n a, o => addAux e n (add a o)
instance : Add ONote :=
⟨add⟩
@[simp]
| Mathlib/SetTheory/Ordinal/Notation.lean | 373 | 379 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra... | end Semiring
section CommSemiring
variable [CommSemiring R] [AddMonoid ι] [SetLike σ R] [AddSubmonoidClass σ R]
| Mathlib/Algebra/DirectSum/Internal.lean | 346 | 349 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... |
@[simp]
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 146 | 147 |
/-
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.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Ring.Subsemiring.Basic
import Mathlib.Algebra.Ring.Opposite
/-!
# Subsemiring of opposite semirings
For e... | theorem unop_sInf (S : Set (Subsemiring Rᵐᵒᵖ)) : (sInf S).unop = sInf (.op ⁻¹' S) :=
opEquiv.symm.map_sInf_eq_sInf_symm_preimage _
theorem op_iSup (S : ι → Subsemiring R) : (iSup S).op = ⨆ i, (S i).op := opEquiv.map_iSup _
| Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean | 129 | 132 |
/-
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.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
/-!
# Properties of cyclic permutations constructed... | rw [toCycle, key]
simp [hx]
theorem nodup_toCycle (f : Perm α) (hf : IsCycle f) : (toCycle f hf).Nodup := by
obtain ⟨x, hx, -⟩ := id hf
simpa [toCycle_eq_toList f hf x hx] using nodup_toList _ _
theorem nontrivial_toCycle (f : Perm α) (hf : IsCycle f) : (toCycle f hf).Nontrivial := by
obtain ⟨x, hx, -⟩ := i... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 372 | 382 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | Mathlib/Data/Bool/Basic.lean | 115 | 115 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,024 | 1,033 | |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,012 | 1,014 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | lift y to ℕ using dom_of_le_natCast ((le_add_left le_rfl).trans_eq h)
rw [← Nat.cast_add, natCast_inj] at h
rw [get_natCast, natCast_inj, eq_tsub_of_add_eq h]
protected theorem add_lt_add_right {x y z : PartENat} (h : x < y) (hz : z ≠ ⊤) : x + z < y + z := by
rcases ne_top_iff.mp (ne_top_of_lt h) with ⟨m, rfl⟩... | Mathlib/Data/Nat/PartENat.lean | 417 | 424 |
/-
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.Group.Action.End
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.Group.Subgroup.MulOppositeLemmas
import Mathlib.Algebra.Gr... | @[to_additive (attr := simp)]
lemma mul_subgroupClosure (hs : s.Nonempty) : s * closure s = closure s := by
rw [← smul_eq_mul, ← Set.iUnion_smul_set]
| Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 66 | 68 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | head?_cons]⟩,
fun ⟨b, hb⟩ n => by rw [hb, rotate_replicate]⟩
theorem rotate_one_eq_self_iff_eq_replicate [Nonempty α] {l : List α} :
| Mathlib/Data/List/Rotate.lean | 261 | 264 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | theorem U_neg_two : U R (-2) = -1 := by
simpa [zero_sub, Int.reduceNeg, U_neg_one, mul_zero, U_zero] using U_sub_two R 0
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 203 | 204 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
theorem ncard_le_one (hs : s.Finite := by toFinite_tac) :
s.ncard ≤ 1 ↔ ∀ a ∈ s, ∀ b ∈ s, a = b := by
simp_rw [ncard_eq_toFinset_card _ hs, Finset.card_le_one, Finite.mem_toFinset]
@[simp] theorem ncard_le_one_iff_subsingleton [Finite s] :
s.ncard ≤ 1 ↔ s.Subsingleton :=
ncard_le_one <| inferInstanceAs (F... | Mathlib/Data/Set/Card.lean | 1,017 | 1,028 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasBasis p s) (i : ι') (hi : p i) (h'i : s i = ∅) :
liminf f v = sSup univ := by
| Mathlib/Order/LiminfLimsup.lean | 284 | 285 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | have h :
∀ B B' : M₁ →ₗ[R] M₂ →ₗ[R] R,
B = B' ↔ LinearMap.toMatrix₂ b₁ b₂ B = LinearMap.toMatrix₂ b₁ b₂ B' := by
intro B B'
constructor <;> intro h
· rw [h]
· exact (LinearMap.toMatrix₂ b₁ b₂).injective h
simp_rw [h, LinearMap.toMatrix₂_comp b₂ b₂, LinearMap.toMatrix₂_compl₂ b₁ b₁,
Lin... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 528 | 542 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
| Mathlib/RingTheory/Coprime/Lemmas.lean | 50 | 54 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn, Sabbir Rahman
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algeb... | Mathlib/Algebra/Order/Field/Power.lean | 181 | 182 | |
/-
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.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | Mathlib/Data/List/OfFn.lean | 250 | 253 | |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | iff both `p` and `q` occur frequently along the corresponding filters. -/
theorem frequently_prod_and {p : α → Prop} {q : β → Prop} :
| Mathlib/Order/Filter/Prod.lean | 409 | 410 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | change IsLowerSet ((f : α ≃ β) '' s)
rw [Set.image_equiv_eq_preimage_symm]
| Mathlib/Order/UpperLower/Basic.lean | 207 | 208 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | simpa [mul_comm] using integral_comp_smul_deriv_Ioi hf hft hff' hg_cont hg1 hg2'
/-- Substitution `y = x ^ p` in integrals over `Ioi 0` -/
theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
(∫ x in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y := by
let S := Ioi (0 : ℝ)
have a1 :... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,068 | 1,079 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an element `x` of an `A... | apply Set.ext_iff.mp Subtype.range_val_subtype
end
| Mathlib/FieldTheory/Minpoly/Basic.lean | 218 | 221 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | (strictMono_rpow_of_pos hz).lt_iff_lt
theorem le_rpow_inv_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y := by
nth_rw 1 [← rpow_one x]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 743 | 746 |
/-
Copyright (c) 2023 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Order.Filter.Germ.Basic
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.Normed.Module.Basic
/-! ... | (preconnectedSpace_iff_univ.mp i) (by trivial) (by trivial)
lemma eq_of_germ_isConstant_on {s : Set X} (h : ∀ x ∈ s, (f : Germ (𝓝 x) Y).IsConstant)
(hs : IsPreconnected s) {x' : X} (x_in : x ∈ s) (x'_in : x' ∈ s) : f x = f x' := by
| Mathlib/Topology/Germ.lean | 164 | 167 |
/-
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.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | exact ⟨i, rfl⟩
· rintro ⟨i, h⟩
rw [← (ofHoms f).arrow_mk_mem_toSet_iff, h, arrow_mk_mem_toSet_iff]
constructor
@[simp]
lemma ofHoms_homFamily (P : MorphismProperty C) : ofHoms P.homFamily = P := by
ext _ _ f
constructor
· intro hf
rw [ofHoms_iff] at hf
| Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 158 | 168 |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Semirings and ... | Mathlib/Algebra/Ring/Semiconj.lean | 95 | 97 | |
/-
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, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | simpa using ((hg.restrictScalars 𝕜).comp x hh hL).hasDerivAtFilter
theorem HasDerivAtFilter.scomp_of_eq (hg : HasDerivAtFilter g₁ g₁' y L')
(hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Tendsto h L L') :
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 74 | 77 |
/-
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 | 733 | 745 | |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 374 | 379 | |
/-
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.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,258 | 1,262 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | variable [IsTopologicalAddGroup E] {s t : Set E}
theorem IsVonNBounded.of_sub_right (hst : IsVonNBounded 𝕜 (s - t)) (hs : s.Nonempty) :
| Mathlib/Analysis/LocallyConvex/Bounded.lean | 324 | 326 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | theorem image2_empty_left : image2 f ∅ t = ∅ :=
ext <| by simp
| Mathlib/Data/Set/NAry.lean | 107 | 108 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | a significant performance hit when `map_one` fails to apply.
| Mathlib/Algebra/Group/Hom/Defs.lean | 209 | 209 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | Mathlib/Data/List/Lattice.lean | 234 | 250 | |
/-
Copyright (c) 2020 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.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | (measurable_derivWithin_Ici f).aemeasurable
theorem aestronglyMeasurable_derivWithin_Ici (μ : Measure ℝ) :
AEStronglyMeasurable (fun x => derivWithin f (Ici x) x) μ :=
(stronglyMeasurable_derivWithin_Ici f).aestronglyMeasurable
| Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 743 | 747 |
/-
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.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | apply totalAux.ιMapObj_D₁
@[reassoc (attr := simp)]
lemma ι_D₂ (i₁₂ i₁₂' : I₁₂) (i₁ : I₁) (i₂ : I₂)
(h : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = i₁₂) :
K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ K.D₂ c₁₂ i₁₂ i₁₂' =
| Mathlib/Algebra/Homology/TotalComplex.lean | 295 | 300 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | x ∈ (Ideal.span {(p : ℤ_[p]) ^ n} : Ideal ℤ_[p]) ↔ n ≤ x.valuation := by
rw [Ideal.mem_span_singleton]
constructor
· rintro ⟨c, rfl⟩
| Mathlib/NumberTheory/Padics/PadicIntegers.lean | 384 | 387 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | Mathlib/Data/List/Zip.lean | 235 | 236 | |
/-
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.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... |
@[to_additive (attr := simp)]
theorem noncommProd_singleton (a : α) (f : α → β) :
noncommProd ({a} : Finset α) f
| Mathlib/Data/Finset/NoncommProd.lean | 301 | 304 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Commutator.Finite
import Mathlib.GroupTheory.Transfer
import Mathlib.Algebra.Group.Pointwise.Fi... | obtain ⟨g, hg, r, hr, rfl⟩ :=
show h ∈ _ from eq_top_iff.mp (closure_mul_image_mul_eq_top hR hR1 hS) (mem_top h)
suffices (⟨r, hr⟩ : R) = (⟨1, hR1⟩ : R) by
simpa only [show r = 1 from Subtype.ext_iff.mp this, mul_one]
apply (isComplement_iff_existsUnique_mul_inv_mem.mp hR r).unique
· rw [Subtype.coe_mk,... | Mathlib/GroupTheory/Schreier.lean | 98 | 113 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | rw [nodup_cons] at hl
simp only [mem_cons, map, ihl hl.2]
by_cases H : hd = x
| Mathlib/Data/List/Nodup.lean | 362 | 364 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,999 | 2,002 | |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Algebra.Field.Basic
/-!
# `norm_num` plugins for `Rat.cast` and `⁻¹`.
-/
va... | theorem isRat_mkRat : {a na n : ℤ} → {b nb d : ℕ} → IsInt a na → IsNat b nb →
IsRat (na / nb : ℚ) n d → IsRat (mkRat a b) n d
| _, _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, ⟨_, h⟩ => by rw [Rat.mkRat_eq_div]; exact ⟨_, h⟩
| Mathlib/Tactic/NormNum/Inv.lean | 56 | 58 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | (Primrec.fst (α := (List (List (Option ℕ)) × ℕ) × ℕ)
(β := Code × Code × Option ℕ × Option ℕ))
have k := k.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
have n := n.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
have cf := Primrec.fst.comp (Primrec.snd (α := (List (L... | Mathlib/Computability/PartrecCode.lean | 844 | 944 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | -/
| Mathlib/Data/Real/GoldenRatio.lean | 117 | 119 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,700 | 2,703 | |
/-
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.Algebra.Group.Fin.Basic
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Tactic.Abel
/-!
# Circulant matrices
This file contains the defini... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 192 | 194 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,681 | 1,682 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... |
/-! ### Declarations about `embDomain` -/
| Mathlib/Data/Finsupp/Defs.lean | 356 | 358 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | (h : ∀ r : σ → R, eval r F = eval r G) (hnR : n ≤ #R) :
F = G := by
rw [← sub_eq_zero]
apply eq_zero_of_forall_eval_eq_zero_of_le_card (hF.sub hG) _ hnR
simpa [sub_eq_zero] using h
/-- See `MvPolynomial.IsHomogeneous.eq_zero_of_forall_eval_eq_zero_of_le_card`
for a version that assumes `n ≤ #R`. -/
lemma... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 409 | 422 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Formal power se... | ext i; simp +contextual [coeff_trunc, Polynomial.coeff_C]
lemma trunc_one_X : trunc (R := R) 1 X = 0 := by simp
@[simp]
lemma trunc_C_mul (n : ℕ) (r : R) (f : R⟦X⟧) : trunc n (C R r * f) = .C r * trunc n f := by
ext i; simp [coeff_trunc]
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 134 | 140 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Floris van Doorn
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# `norm_num` plugin for ... | ⟨by simp⟩
/-- Evaluate a big operator applied to a finset by repeating `proveEmptyOrCons` until
| Mathlib/Tactic/NormNum/BigOperators.lean | 326 | 328 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | rw [← add_eq_zero_iff_eq_neg, ← two_nsmul, two_nsmul_eq_zero_iff]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 173 | 173 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic
import Mathlib.Topology.Order... | that belongs to `(1, ε)`. -/
@[to_additive "An additive subgroup of an archimedean linear ordered additive commutative group with
order topology is dense provided that for all positive `ε` there exists a positive element of the
subgroup that is less than `ε`."]
theorem dense_of_not_isolated_one (S : Subgroup G) (hS : ∀... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 42 | 53 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Group.Pointwise.Set.Scalar
import Mathlib.Algebra.Ring.Subsemiring.Basic
import Mathlib.R... | variable (M)
variable [IsLocalization M S]
/-- Each element `a : S` has an `M`-multiple which is an integer.
| Mathlib/RingTheory/Localization/Integer.lean | 63 | 66 |
/-
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.Complex.Log
/-! # Power funct... | -- let nc ← mk_instance_cache q(ℕ)
-- let (icβ, nc, b', hb) ← prove_nat_uncast icβ nc b
| Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 267 | 268 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.GameAdd
import Mathlib.Order.RelIso.Set
import Mathlib.SetTheory.ZFC.Basic
/-!
# Von Neumann ordinals
This file works t... | rw [← sUnion_pair]
apply IsTransitive.sUnion'
intro
rw [mem_pair]
| Mathlib/SetTheory/ZFC/Ordinal.lean | 71 | 74 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathlib.Algebra.Polyno... | Mathlib/Algebra/Polynomial/Taylor.lean | 137 | 142 | |
/-
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.... | (hi := Fin.ne_zero_of_lt h) : p.predAbove i = i.pred hi := dif_pos h
lemma predAbove_of_succ_le (p : Fin n) (i : Fin (n + 1)) (h : succ p ≤ i)
| Mathlib/Data/Fin/Basic.lean | 1,188 | 1,190 |
/-
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.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... |
@[simp]
theorem equivAddCircle_symm_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
| Mathlib/Topology/Instances/AddCircle.lean | 323 | 325 |
/-
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.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
/-!
# Lemmas about images of inte... | theorem image_Ioc (e : α ≃o β) (a b : α) : e '' Ioc a b = Ioc (e a) (e b) := by
rw [e.image_eq_preimage, e.symm.preimage_Ioc, e.symm_symm]
| Mathlib/Order/Interval/Set/OrderIso.lean | 78 | 79 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... |
theorem add_divOf (x y : k[G]) (g : G) : (x + y) /ᵒᶠ g = x /ᵒᶠ g + y /ᵒᶠ g :=
map_add (Finsupp.comapDomain.addMonoidHom _) _ _
| Mathlib/Algebra/MonoidAlgebra/Division.lean | 77 | 79 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,281 | 1,284 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,492 | 1,494 | |
/-
Copyright (c) 2022 Joseph Hua. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta, Johan Commelin, Reid Barton, Robert Y. Lewis, Joseph Hua
-/
import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
import Mathlib.CategoryTheory.Functor.EpiMono... | /-
```
str
V₀ -----> F V₀
| Mathlib/CategoryTheory/Endofunctor/Algebra.lean | 238 | 241 |
/-
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.Geometry.RingedSpace.PresheafedSpace
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Topology.Sheaves.Limits
import Mathlib.CategoryThe... |
instance : PreservesColimitsOfShape J (PresheafedSpace.forget.{u, v, v} C) :=
⟨fun {F} => preservesColimit_of_preserves_colimit_cocone (colimitCoconeIsColimit F) <| by
apply IsColimit.ofIsoColimit (colimit.isColimit _)
fapply Cocones.ext
· rfl
· intro j
simp⟩
/-- When `C` has limits, the categ... | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | 266 | 286 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | rw [← whiskerLeft_comp, IsIso.hom_inv_id, whiskerLeft_id]
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 302 | 304 |
/-
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.Integral.IntervalIntegral.Periodic
deprecated_module (since := "2025-04-13")
| Mathlib/MeasureTheory/Integral/Periodic.lean | 267 | 274 | |
/-
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.sub (hf : HasDerivWithinAt f f' s x)
(hg : HasDerivWithinAt g g' s x) : HasDerivWithinAt (fun x => f x - g x) (f' - g') s x :=
| Mathlib/Analysis/Calculus/Deriv/Add.lean | 297 | 299 |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Dynamics.PeriodicPts.Lemmas
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This modul... | theorem period_eq_one_iff {m : M} {a : α} : period m a = 1 ↔ m • a = a :=
⟨fun eq_one => pow_one m ▸ eq_one ▸ pow_period_smul m a,
fun fixed => le_antisymm
(period_le_of_fixed one_pos (by simpa))
(period_pos_of_fixed one_pos (by simpa))⟩
| Mathlib/GroupTheory/GroupAction/Period.lean | 49 | 53 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
/-... | @[deprecated (since := "2025-01-26")]
alias isLocallySurjective_iff_imagePresheaf_sheafify_eq_top :=
isLocallySurjective_iff_range_sheafify_eq_top
attribute [local instance] Types.instFunLike Types.instConcreteCategory in
| Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 101 | 105 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | @[simp]
theorem embDomain_add (f : α ↪ β) (v w : α →₀ M) :
embDomain f (v + w) = embDomain f v + embDomain f w :=
(embDomain.addMonoidHom f).map_add v w
| Mathlib/Data/Finsupp/Defs.lean | 555 | 558 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,979 | 2,981 | |
/-
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.Functor.ReflectsIso.Basic
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# Morphism properties that are inverted by a functor
In this ... | Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 161 | 163 | |
/-
Copyright (c) 2021 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Homotopy.Path
/-!
# Product of homotopies
In this file, we introduce definitions for the produ... | Mathlib/Topology/Homotopy/Product.lean | 238 | 243 | |
/-
Copyright (c) 2023 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.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
/-!
# Derivatives of `x ↦ x⁻¹` and `f x / g x`
In this... | · simp only [div_eq_mul_inv, (· ∘ ·)]
· field_simp
ring
| Mathlib/Analysis/Calculus/Deriv/Inv.lean | 132 | 135 |
/-
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.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | ### The circular order structure on `α ⧸ AddSubgroup.zmultiples p`
-/
| Mathlib/Algebra/Order/ToIntervalMod.lean | 713 | 716 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-20")
| Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 152 | 169 | |
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
/-!
# Vieta's Formula
The main result is `Multiset.prod_X_add_C_eq_sum_... |
end Semiring
| Mathlib/RingTheory/Polynomial/Vieta.lean | 75 | 77 |
/-
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.Order.BooleanAlgebra
import Mathlib.Order.Hom.Lattice
/-!
# Adding complements to a generalized Boolean algebra
This file embeds any generalized Boolean ... | @[simp] lemma comp_le_comp : comp a ≤ comp b ↔ b ≤ a := ⟨by rintro ⟨_⟩; assumption, LE.comp⟩
| Mathlib/Order/Booleanisation.lean | 137 | 137 |
/-
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 | 1,155 | 1,159 | |
/-
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.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
/-!
# Construction of homotopies for the Dold-Kan correspondence
(The general str... | rw [nullHomotopicMap'_comp, comp_nullHomotopicMap']
congr
ext n m hnm
simp only [alternatingFaceMapComplex_map_f, hσ'_naturality]
/-- The maps `hσ' q n m hnm` are compatible with the application of additive functors. -/
theorem map_hσ' {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 156 | 166 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | theorem card_le_four : #{a, b, c, d} ≤ 4 :=
(card_insert_le _ _).trans (Nat.succ_le_succ card_le_three)
theorem card_le_five : #{a, b, c, d, e} ≤ 5 :=
(card_insert_le _ _).trans (Nat.succ_le_succ card_le_four)
| Mathlib/Data/Finset/Card.lean | 115 | 119 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,220 | 1,221 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.... | rw [hμ, comp_add, map_add, hg, add_zero, this, LinearMap.map_smul, smul_eq_mul]
-- This result requires `Mathlib.RingTheory.TensorProduct.Free`. Maybe it should move elsewhere?
@[simp]
lemma trace_baseChange [Module.Free R M] [Module.Finite R M]
(f : M →ₗ[R] M) (A : Type*) [CommRing A] [Algebra R A] :
| Mathlib/LinearAlgebra/Trace.lean | 316 | 321 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | @[simp]
theorem modEq_zero : a ≡ b [PMOD 0] ↔ a = b := by simp [ModEq, sub_eq_zero, eq_comm]
| Mathlib/Algebra/ModEq.lean | 89 | 90 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,034 | 1,035 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Batteries.Data.List.Perm
import Mathlib.Data.List.Nodup
/-!
# Lists of elements of `Fin n`... |
theorem ofFn_id (n) : ofFn id = finRange n :=
rfl
| Mathlib/Data/List/FinRange.lean | 87 | 90 |
/-
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.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.RingTheory.LocalProperties.Basic
import Mathlib.RingTheory.RingHom.Locally
/-!
# Properties of morp... | exact hP.pullback_fst_appTop _ (isLocal_ringHomProperty P).respectsIso _ _ H
include Q in
private lemma respects_isOpenImmersion_aux
(hQ : RingHom.StableUnderCompositionWithLocalizationAwaySource Q)
{X Y : Scheme.{u}} [IsAffine Y] {U : Y.Opens}
(f : X ⟶ U.toScheme) (hf : P f) : P (f ≫ U.ι) := by
wlog h... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 525 | 565 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | Mathlib/Algebra/ModEq.lean | 311 | 312 | |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... |
@[reassoc]
theorem congr_hom {F G : C ⥤ D} (h : F = G) {X Y} (f : X ⟶ Y) :
F.map f = eqToHom (congr_obj h X) ≫ G.map f ≫ eqToHom (congr_obj h Y).symm := by
| Mathlib/CategoryTheory/EqToHom.lean | 265 | 268 |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... | theorem comp_compQuasiMeasurePreserving
{β : Type*} [MeasurableSpace β] {ν} (g : γ → δ) (hg : Continuous g)
(f : β →ₘ[ν] γ) {φ : α → β} (hφ : Measure.QuasiMeasurePreserving φ μ ν) :
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 282 | 284 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | theorem natCast_get {x : PartENat} (h : x.Dom) : (x.get h : PartENat) = x := by
exact Part.ext' (iff_of_true trivial h) fun _ _ => rfl
| Mathlib/Data/Nat/PartENat.lean | 165 | 166 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Kim Morrison, Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.Cat... | ToType (pullback f₁ f₂) :=
(pullbackEquiv f₁ f₂).symm ⟨⟨x₁, x₂⟩, h⟩
lemma pullbackMk_surjective (x : ToType (pullback f₁ f₂)) :
| Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 202 | 205 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | @[deprecated "Deprecated without replacement." (since := "2025-01-29")]
def foldlIdxSpec (f : ℕ → α → β → α) (a : α) (bs : List β) (start : ℕ) : α :=
foldl (fun a p ↦ f p.fst a p.snd) a <| enumFrom start bs
set_option linter.deprecated false in
@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
... | Mathlib/Data/List/Indexes.lean | 134 | 140 |
/-
Copyright (c) 2022 Yaël Dillies, Ella Yu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Ella Yu
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.Prod
import Mathlib.Algebra.Group.Poin... | @[to_additive (attr := simp)]
lemma mulEnergy_univ_left : Eₘ[univ, t] = Fintype.card α * t.card ^ 2 := by
simp only [mulEnergy, univ_product_univ, Fintype.card, sq, ← card_product]
let f : α × α × α → (α × α) × α × α := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)
have : (↑((univ : Finset α) ×ˢ t ×ˢ t) : Set (α × α... | Mathlib/Combinatorics/Additive/Energy.lean | 172 | 177 |
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