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 |
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/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Junyan Xu, Sophie Morel
-/
import Mathlib.CategoryTheory.Limits.Creates
import Mathlib.CategoryTheory.Limits.Types.Limits
import Mathlib.CategoryTheory.Limits.Types... | Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean | 101 | 107 | |
/-
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.CharP.Lemmas
import Mathlib.GroupTheory.OrderOfElement
/-!
# Lemmas about rings of characteristic two
This file contains results about `CharP R 2`,... | cast_eq_zero_iff n := by
| Mathlib/Algebra/CharP/Two.lean | 33 | 33 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | theorem unifIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {f : ι → α → β}
(hf : ∀ i, MemLp (f i) p μ) : UnifIntegrable f p μ := by
obtain ⟨n, hn⟩ := Finite.exists_equiv_fin ι
intro ε hε
let g : Fin n → α → β := f ∘ hn.some.symm
have hg : ∀ i, MemLp (g i) p μ := fun _ => hf _
obtain ⟨δ, hδ... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 451 | 461 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | theorem eval_eq_smeval : p.eval r = p.smeval r := by
rw [eval_eq_sum, smeval_eq_sum]
rfl
| Mathlib/Algebra/Polynomial/Smeval.lean | 65 | 67 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... |
theorem eq_zero_of_ne_zero_of_mul_right_eq_zero (hx : x ≠ 0) (hxy : y * x = 0) : y = 0 :=
Or.resolve_right (eq_zero_or_eq_zero_of_mul_eq_zero hxy) hx
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 164 | 166 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defines what it means for a... |
theorem cg_iff_countable [Countable (Σ l, L.Functions l)] : CG L M ↔ Countable M := by
rw [cg_def, Substructure.cg_iff_countable, topEquiv.toEquiv.countable_iff]
theorem cg_of_countable [Countable M] : CG L M := by
simp only [cg_def, Substructure.cg_of_countable, topEquiv.toEquiv.countable_iff]
theorem FG.cg (h ... | Mathlib/ModelTheory/FinitelyGenerated.lean | 267 | 274 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... |
variable {f : α → ℚ → ℝ}
/-- `a : α` is a Stieltjes point for `f : α → ℚ → ℝ` if `f a` is monotone with limit 0 at -∞
and 1 at +∞ and satisfies a continuity property. -/
| Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 66 | 70 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.Submodule
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Mid... | dist_self x := by simp
dist_comm x y := by simp only [← neg_vsub_eq_vsub_rev y x, norm_neg]
dist_triangle x y z := by
rw [← vsub_add_vsub_cancel]
| Mathlib/Analysis/Normed/Group/AddTorsor.lean | 179 | 182 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... |
/- TODO: Move to another file? -/
open Submodule
variable {K} {x y : E}
theorem eq_zero_of_inner_left (hK : Dense (K : Set E)) (h : ∀ v : K, ⟪x, v⟫ = 0) : x = 0 := by
have : (⟪x, ·⟫) = 0 := (continuous_const.inner continuous_id).ext_on
hK continuous_const (Subtype.forall.1 h)
simpa using congr_fun this x
th... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 921 | 944 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... |
instance (priority := 100) : PathConnectedSpace 𝕜 := by
letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
infer_instance
/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
| Mathlib/Analysis/Calculus/MeanValue.lean | 413 | 418 |
/-
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... | Mathlib/CategoryTheory/Monoidal/Category.lean | 1,047 | 1,050 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... | | Matrix.vecCons _ n x xs => do
let (elems, n', tail) ← matchVecConsPrefix n xs
return (x :: elems, n', tail)
| Mathlib/Data/Fin/VecNotation.lean | 146 | 148 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Jujian Zhang, Yongle Hu
-/
import Mathlib.Algebra.Colimit.TensorProduct
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.Algebra.DirectSum.Module
import Mathlib.... | let se := (Shrink.linearEquiv.{_, u} P R).symm
have := Module.Finite.equiv se
rw [rTensor_injective_iff_subtype (fun _ _ ↦ (Subtype.ext <| hf <| Subtype.ext_iff.mp ·)) se]
exact (flat_iff R M).mp ‹_› _ (Finite.iff_fg.mp inferInstance)
/-- If `M` is a flat module, then `𝟙 M ⊗ f` is injective for all injective ... | Mathlib/RingTheory/Flat/Basic.lean | 123 | 139 |
/-
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, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | theorem blockTriangular_stdBasisMatrix' {i j : m} (hij : b j ≤ b i) (c : R) :
BlockTriangular (stdBasisMatrix i j c) (toDual ∘ b) :=
blockTriangular_stdBasisMatrix (by exact toDual_le_toDual.mpr hij) _
end Zero
variable [CommRing R] [DecidableEq m]
| Mathlib/LinearAlgebra/Matrix/Block.lean | 152 | 159 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
/-!
# Quadratic reciprocity.
## Main r... | rcases eq_or_ne p q with h | h
· subst p
rw [(eq_zero_iff q q).mpr (mod_cast natCast_self q), mul_zero]
· have qr := congr_arg (· * legendreSym p q) (quadratic_reciprocity hp hq h)
have : ((q : ℤ) : ZMod p) ≠ 0 := mod_cast prime_ne_zero p q h
simpa only [mul_assoc, ← pow_two, sq_one p this, mul_one] u... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 121 | 133 |
/-
Copyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Bhavik Mehta, Kyle Miller
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Set.Finite.Basic
/-!
#... | exact ⟨isEmptyElim, isEmptyElim, isEmptyElim⟩
· have ih' : ∀ (ι' : Type u) [Fintype ι'] (t' : ι' → Finset α), Fintype.card ι' ≤ n →
(∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →
∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x := by
intro ι' _ _ hι' ht'
exact ih _ (Nat.lt_succ_of_le... | Mathlib/Combinatorics/Hall/Finite.lean | 226 | 243 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | theorem iSupIndep_def' : iSupIndep t ↔ ∀ i, Disjoint (t i) (sSup (t '' { j | j ≠ i })) := by
simp_rw [sSup_image]
rfl
@[deprecated (since := "2024-11-24")] alias CompleteLattice.independent_def' := iSupIndep_def'
| Mathlib/Order/SupIndep.lean | 328 | 332 |
/-
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.Algebra.Defs
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Math... | universalHom f fS hf (r /ₒ s) = ((fS s)⁻¹ : Units T) * f r :=
rfl
theorem universalHom_commutes {r : R} : universalHom f fS hf (numeratorHom r) = f r := by
simp [numeratorHom_apply, universalHom_apply]
theorem universalHom_unique (φ : R[S⁻¹] →+* T) (huniv : ∀ r : R, φ (numeratorHom r) = f r) :
φ = univers... | Mathlib/RingTheory/OreLocalization/Ring.lean | 166 | 173 |
/-
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.Analysis.NormedSpace.Multilinear.Curry
/-!
# Formal multilinear series
In this file we define `FormalMultilinearSeries 𝕜 E F` to be a family o... | congr with ⟨i, hi⟩
exact h2 i hi hi
lemma congr_zero (p : FormalMultilinearSeries 𝕜 E F) {k l : ℕ} (h : k = l) (h' : p k = 0) :
p l = 0 := by
subst h; exact h'
| Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 124 | 129 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... |
end Constructions
/-!
### Expressing continuity properties in uniform spaces
We reformulate the various continuity properties of functions taking values in a uniform space
in terms of the uniformity in the target. Since the same lemmas (essentially with the same names)
also exist for metric spaces and emetric spaces... | Mathlib/Topology/UniformSpace/Basic.lean | 892 | 900 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | coe_le_iff.2 <| by
rintro b rfl
apply NNReal.inv_le_of_le_mul
rwa [← coe_mul, ← coe_one, coe_le_coe] at hb
@[simp, norm_cast]
| Mathlib/Data/ENNReal/Inv.lean | 54 | 59 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Finsupp
import Mathlib.Order.Filter.AtTopBot.Defs
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Gra... | map_one' := val_one
map_mul' := val_mul
| Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 449 | 450 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | theorem card_cast_subgroup_card_ne_zero [Ring K] [NoZeroDivisors K] [Nontrivial K]
(G : Subgroup Kˣ) [Fintype G] : (Fintype.card G : K) ≠ 0 := by
let n := Fintype.card G
intro nzero
have ⟨p, char_p⟩ := CharP.exists K
have hd : p ∣ n := (CharP.cast_eq_zero_iff K p n).mp nzero
cases CharP.char_is_prime_or_z... | Mathlib/FieldTheory/Finite/Basic.lean | 114 | 138 |
/-
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... | · refine ⟨x'.lim, fun n => ?_⟩
have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := zpow_pos (mod_cast hp.out.pos) _
| Mathlib/NumberTheory/Padics/PadicIntegers.lean | 472 | 473 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem ext ⦃inst₁ inst₂ : NonUnitalNonAssocCommRing R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂ :=
| Mathlib/Algebra/Ring/Ext.lean | 387 | 392 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
/-!
# Ordered monoids
This file provides the defin... | @[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
| Mathlib/Algebra/Order/Monoid/Defs.lean | 172 | 173 |
/-
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 encard_mono {α : Type*} : Monotone (encard : Set α → ℕ∞) :=
fun _ _ ↦ encard_le_encard
theorem encard_diff_add_encard_of_subset (h : s ⊆ t) : (t \ s).encard + s.encard = t.encard := by
| Mathlib/Data/Set/Card.lean | 169 | 172 |
/-
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.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.HomCongr
/-!
# Restricting adjunctions
`Adjunction.restrictFullyFaithful` shows that an adj... |
@[simp, reassoc]
lemma map_restrictFullyFaithful_counit_app (X : D) :
iD.map ((adj.restrictFullyFaithful hiC hiD comm1 comm2).counit.app X) =
comm1.inv.app (R.obj X) ≫ L'.map (comm2.inv.app X) ≫ adj.counit.app (iD.obj X) := by
dsimp [restrictFullyFaithful]
| Mathlib/CategoryTheory/Adjunction/Restrict.lean | 62 | 67 |
/-
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, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... |
variable (hu : IsUnit x)
| Mathlib/Algebra/Divisibility/Units.lean | 180 | 182 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | W.ψ (m + 2) * W.ψ m ^ 3 - W.ψ (m - 1) * W.ψ (m + 1) ^ 3 :=
normEDS_odd ..
lemma Affine.CoordinateRing.mk_ψ (n : ℤ) : mk W (W.ψ n) = mk W (W.Ψ n) := by
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 484 | 487 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 347 | 349 | |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li, Joachim Breitner
-/
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.SuccPred.WithBot
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.... |
variable {α β : Type*}
variable [Preorder α] [Preorder β]
@[simp] lemma height_toDual (x : α) : height (OrderDual.toDual x) = coheight x := rfl
| Mathlib/Order/KrullDimension.lean | 109 | 114 |
/-
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-... |
@[simp]
| Mathlib/Order/WellFoundedSet.lean | 449 | 450 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 2,094 | 2,116 | |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | rwa [← separatingRight_iff_flip_ker_eq_bot]
lemma disjoint_ker_of_nondegenerate_restrict {B : M →ₗ[R] M →ₗ[R] M₁} {W : Submodule R M}
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 719 | 721 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Int.Sqrt
/-!... |
end Rat
| Mathlib/Data/Rat/Sqrt.lean | 51 | 52 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | use fun (s₁, s₂) s => s₂ = s ∧ s₁ = s
simp_all
/-- If `f` takes a pair of states, but always returns the same value for both elements of the
pair, then we can simplify to just a single element of state.
-/
@[simp]
theorem mapAccumr₂_redundant_pair (f : α → β → (σ × σ) → (σ × σ) × γ)
| Mathlib/Data/Vector/MapLemmas.lean | 320 | 327 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,174 | 1,180 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | simp_rw [h_eq, measureReal_def]
rw [← ENNReal.tsum_toReal_eq (fun _ ↦ measure_ne_top _ _)]
congr
rw [iUnion_prod_const, measure_iUnion]
· exact hf_disj.mono fun _ _ h ↦ h.set_prod_left _ _
· exact fun i ↦ (hf i).prod hs
lemma setLIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α)... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 567 | 605 |
/-
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... | f ∘ g⁻¹ = (f ∘ g)⁻¹ := by ext; simp
/-- Group homomorphisms preserve division. -/
@[to_additive "Additive group homomorphisms preserve subtraction."]
| Mathlib/Algebra/Group/Hom/Defs.lean | 431 | 434 |
/-
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... | csInf_le hf h
theorem le_limsInf_of_le {f : Filter α} {a}
(hf : f.IsCobounded (· ≥ ·) := by isBoundedDefault)
| Mathlib/Order/LiminfLimsup.lean | 129 | 132 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | let r₂ := f₂ x y s₂
let r₁ := f₁ x r₂.snd s₁
((r₁.fst, r₂.fst), r₁.snd)
)
xs ys (s₁, s₂)
(m.fst.fst, m.snd) := by
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
@[simp]
theorem mapAccumr₂_mapAccumr₂_right_rig... | Mathlib/Data/Vector/MapLemmas.lean | 145 | 154 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAl... | Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 527 | 528 | |
/-
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 Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,267 | 2,268 | |
/-
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 | 1,610 | 1,617 | |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
/-!
# Conditional expectation of indicator functions
This file proves some results about the conditi... |
theorem condExp_restrict_ae_eq_restrict (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
(hs_m : MeasurableSet[m] s) (hf_int : Integrable f μ) :
(μ.restrict s)[f|m] =ᵐ[μ.restrict s] μ[f|m] := by
have : SigmaFinite ((μ.restrict s).trim hm) := by rw [← restrict_trim hm _ hs_m]; infer_instance
rw [ae_eq_restrict_iff_i... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 115 | 140 |
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in measure
We define con... | refine strictMono_nat_of_lt_succ fun n => ?_
rw [seqTendstoAeSeq_succ]
exact lt_of_lt_of_le (lt_add_one <| seqTendstoAeSeq hfg n) (le_max_right _ _)
end ExistsSeqTendstoAe
/-- If `f` is a sequence of functions which converges in measure to `g`, then there exists a
subsequence of `f` which converges a.e. to `g`.... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 189 | 234 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 496 | 497 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | theorem sdiff_sdiff_left' : (x \ y) \ z = x \ y ⊓ x \ z := by rw [sdiff_sdiff_left, sdiff_sup]
theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) :=
calc
z \ (x \ y ⊔ y \ x) = (z \ x ⊔ z ⊓ x ⊓ y) ⊓ (z \ y ⊔ z ⊓ y ⊓ x) := by
| Mathlib/Order/BooleanAlgebra.lean | 371 | 375 |
/-
Copyright (c) 2019 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... | gcongr; exacts [hf.one_sub_K_pos.le, hf.toLipschitzWith.dist_iterate_succ_le_geometric x n]
theorem tendsto_iterate_fixedPoint (x) :
Tendsto (fun n ↦ f^[n] x) atTop (𝓝 <| fixedPoint f hf) := by
convert tendsto_iterate_efixedPoint hf (edist_ne_top x _)
refine (fixedPoint_unique _ ?_).symm
apply efixedP... | Mathlib/Topology/MetricSpace/Contracting.lean | 293 | 299 |
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... | simp
theorem map_comp_apply (f : M →ₗ[R] N) (g : N →ₗ[R] P) (x : AdicCompletion I M) :
map I g (map I f x) = map I (g ∘ₗ f) x := by
| Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 178 | 181 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... | isoWhiskerLeft _ (shiftFunctorAdd C a₂ a₃) := by
ext X
simpa [shiftFunctorAdd'_eq_shiftFunctorAdd]
using NatTrans.congr_app (congr_arg Iso.hom
(shiftFunctorAdd'_assoc C a₁ a₂ a₃ _ _ _ rfl rfl rfl)) X
variable {C}
lemma shiftFunctorAdd'_zero_add_hom_app (a : A) (X : C) :
(shiftFunctorAdd' C 0 a... | Mathlib/CategoryTheory/Shift/Basic.lean | 261 | 277 |
/-
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 | 1,049 | 1,052 | |
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
/-!
# Strict bicategories
A bicategory is called `Strict` if the left unitors, the ri... | Mathlib/CategoryTheory/Bicategory/Strict.lean | 85 | 88 | |
/-
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.Lie.OfAssociative
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
/-!
# The Lie algebra `... | lemma lie_lie_smul_f (t : IsSl2Triple h e f) : ⁅h, f⁆ = -((2 : R) • f) := by
simp [t.lie_h_f_nsmul, two_smul]
| Mathlib/Algebra/Lie/Sl2.lean | 59 | 60 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.Probability.Process.Filtration
/-!
# Filtration built from the finite partitions of a coun... | lemma iSup_partitionFiltration (ht : ∀ n, MeasurableSet (t n))
(ht_range : generateFrom (Set.range t) = m) :
⨆ n, partitionFiltration ht n = m := by
rw [iSup_partitionFiltration_eq_generateFrom_range ht, ht_range]
| Mathlib/Probability/Process/PartitionFiltration.lean | 91 | 94 |
/-
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.Complex
/-!
# The `arctan` function.
Inequalit... | rw [arccos, eq_comm]
refine arctan_eq_of_tan_eq ?_ ⟨?_, ?_⟩
· rw_mod_cast [tan_pi_div_two_sub, tan_arcsin, inv_div]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 187 | 189 |
/-
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 ... | Mathlib/Order/UpperLower/Basic.lean | 1,316 | 1,318 | |
/-
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.... | exact LinearMap.range_eq_bot.2 rfl
instance : FunLike (twoCoboundaries A) (G × G) A := ⟨Subtype.val, Subtype.val_injective⟩
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 405 | 407 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 1,013 | 1,018 | |
/-
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 | 1,075 | 1,079 | |
/-
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 | 1,659 | 1,660 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Module.Submodule.Pointwise
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Non... | use i
suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i) by
rw [h]
assumption
simp only [image_add_left, neg_add_eq_sub]
ext b
simp
· rintro ⟨i, hi⟩
use B i
constructor
· use i
· rw [image_subset_iff]
rintro b b_in
... | Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | 153 | 175 |
/-
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.MFDeriv.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.Basic
/-!
# Unique derivative sets in manifolds
... | (hf : MDifferentiableOn I I' f s) (hd : ∀ x ∈ s, DenseRange (mfderivWithin I I' f s x)) :
UniqueMDiffOn I' (f '' s) :=
hs.image_denseRange' (fun x hx ↦ (hf x hx).hasMFDerivWithinAt) hd
protected theorem UniqueMDiffWithinAt.preimage_partialHomeomorph (hs : UniqueMDiffWithinAt I s x)
{e : PartialHomeomorph... | Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean | 67 | 73 |
/-
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
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... | ext fun _ _ => rfl
section Zero
variable [Zero α] [Zero β]
| Mathlib/Data/Matrix/Block.lean | 474 | 479 |
/-
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_one_sub' (hx : 0 ≤ x) (h : 1 - y ≠ 0) : x ^ (1 - y) = x / x ^ y := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 445 | 446 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | Irrational (√z) ↔ ¬IsSquare z := by
rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),
Rat.cast_intCast]
theorem irrational_sqrt_intCast_iff {z : ℤ} :
Irrational (√z) ↔ ¬IsSquare z ∧ 0 ≤ z := by
| Mathlib/Data/Real/Irrational.lean | 113 | 118 |
/-
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.Data.Finsupp.Lex
import Mathlib.Algebra.MvPolynomial.Degrees
/-!
# Variables of polynomials
This file establishes man... |
variable [CommSemiring S]
| Mathlib/Algebra/MvPolynomial/Variables.lean | 222 | 223 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Module.Submodule.Map
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.RingTheory.Ideal.Quotient.Defs
/-!
# modular equivalence for submodule
-... | theorem map (hxy : x ≡ y [SMOD U]) (f : M →ₗ[R] N) : f x ≡ f y [SMOD U.map f] :=
(Submodule.Quotient.eq _).2 <| f.map_sub x y ▸ mem_map_of_mem <| (Submodule.Quotient.eq _).1 hxy
theorem comap {f : M →ₗ[R] N} (hxy : f x ≡ f y [SMOD V]) : x ≡ y [SMOD V.comap f] :=
(Submodule.Quotient.eq _).2 <|
show f (x - y) ∈ ... | Mathlib/LinearAlgebra/SModEq.lean | 112 | 117 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | def ppow (f : ArithmeticFunction R) (k : ℕ) : ArithmeticFunction R :=
if h0 : k = 0 then ζ else ⟨fun x ↦ f x ^ k, by simp_rw [map_zero, zero_pow h0]⟩
@[simp]
theorem ppow_zero {f : ArithmeticFunction R} : f.ppow 0 = ζ := by rw [ppow, dif_pos rfl]
@[simp]
theorem ppow_apply {f : ArithmeticFunction R} {k x : ℕ} (kpos... | Mathlib/NumberTheory/ArithmeticFunction.lean | 468 | 475 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... | a * b ≤ b ↔ a ≤ 1 :=
Iff.trans (by rw [one_mul]) (mul_le_mul_iff_right b)
end LE
| Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 474 | 477 |
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Equal and mixed characteristic
In commutative ... | rcases g with ⟨I, ⟨hI_not_top, _⟩⟩
-- Krull's Thm: There exists a prime ideal `M` such that `I ≤ M`.
rcases Ideal.exists_le_maximal I hI_not_top with ⟨M, ⟨hM_max, hM_ge⟩⟩
use M
constructor
· exact hM_max
· cases CharP.exists (R ⧸ M) with
| intro r hr =>
convert hr
have ... | Mathlib/Algebra/CharP/MixedCharZero.lean | 112 | 133 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... |
@[simp]
theorem empty_vecAppend (v : Fin n → α) : vecAppend n.zero_add.symm ![] v = v := by
ext
simp [vecAppend_eq_ite]
@[simp]
theorem cons_vecAppend (ho : o + 1 = m + 1 + n) (x : α) (u : Fin m → α) (v : Fin n → α) :
vecAppend ho (vecCons x u) v = vecCons x (vecAppend (by omega) u v) := by
ext i
simp_rw ... | Mathlib/Data/Fin/VecNotation.lean | 308 | 320 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
/-!
# Nondeterministic Finite Automata
This file contains the definition of a Nondeterministic Finite... | rw [mem_accepts, DFA.mem_accepts]
constructor <;> · exact fun ⟨w, h2, h3⟩ => ⟨w, h3, h2⟩
| Mathlib/Computability/NFA.lean | 107 | 108 |
/-
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.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | Mathlib/RingTheory/PowerSeries/Basic.lean | 245 | 245 | |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.SetTheory.Cardinal.Finite
/-!
# Cardinality of finite types
The cardinality of a finite type `α` is given by `Nat.card α`. This function has
the "junk val... | haveI := Fintype.ofFinite β
simp only [Nat.card_eq_fintype_card, Fintype.card_sum]
theorem card_image_le {s : Set α} [Finite s] (f : α → β) : Nat.card (f '' s) ≤ Nat.card s :=
card_le_of_surjective _ Set.surjective_onto_image
theorem card_range_le [Finite α] (f : α → β) : Nat.card (Set.range f) ≤ Nat.card α :=
... | Mathlib/Data/Finite/Card.lean | 145 | 152 |
/-
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, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... |
variable [CommMonoid α]
| Mathlib/RingTheory/Multiplicity.lean | 524 | 526 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Data.Fintype.Order
import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
/-!
# Extension of field embeddings
`IntermediateField.exis... | ((hτ i).2 ⟨x, hx⟩).trans ((hσ i).2 ⟨x, hx⟩).symm⟩
/-- `σ : L →ₐ[F] K` is an extendible lift ("extendible pair" in [Isaacs1980]) if for every
intermediate field `M` that is finite-dimensional over `L`, `σ` extends to some `M →ₐ[F] K`.
| Mathlib/FieldTheory/Extension.lean | 92 | 95 |
/-
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.... | trace R (M × N) (prodMap f g) = trace R M f + trace R N g := by
have h := LinearMap.ext_iff.1 (trace_prodMap R M N) (f, g)
simp only [coe_comp, Function.comp_apply, prodMap_apply, coprod_apply, id_coe, id,
prodMapLinear_apply] at h
exact h
variable (R M N P)
open TensorProduct Function
theorem trace_te... | Mathlib/LinearAlgebra/Trace.lean | 207 | 221 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Algebra.EuclideanDomain.Field
... | (dickson 1 (1 : R) m).comp (dickson 1 1 n) = (dickson 1 1 n).comp (dickson 1 1 m) := by
rw [← dickson_one_one_mul, mul_comm, dickson_one_one_mul]
theorem dickson_one_one_zmod_p (p : ℕ) [Fact p.Prime] : dickson 1 (1 : ZMod p) p = X ^ p := by
-- Recall that `dickson_one_one_eval_add_inv` characterises `dickson 1... | Mathlib/RingTheory/Polynomial/Dickson.lean | 196 | 262 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,804 | 1,820 | |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Óscar Álvarez
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.List.GetD
import Mathlib.Tactic.Group
/-!
# Reflections, inversions, and inversion sequences... |
theorem leftInvSeq_concat (ω : List B) (i : B) :
lis (ω.concat i) = (lis ω).concat ((π ω) * (s i) * (π ω)⁻¹) := by
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 227 | 229 |
/-
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.Single
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
/-!
# The homology of single complexes
The main definition in this file ... | @[reassoc (attr := simp)]
lemma homologyπ_singleObjHomologySelfIso_hom :
((single C c j).obj A).homologyπ j ≫ (singleObjHomologySelfIso _ _ _).hom =
(singleObjCyclesSelfIso _ _ _).hom := by
simp [singleObjCyclesSelfIso, singleObjHomologySelfIso]
| Mathlib/Algebra/Homology/SingleHomology.lean | 70 | 74 |
/-
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... | theorem exp_mul (x y : ℝ) : exp (x * y) = exp x ^ y := by rw [rpow_def_of_pos (exp_pos _), log_exp]
@[simp, norm_cast]
theorem rpow_intCast (x : ℝ) (n : ℤ) : x ^ (n : ℝ) = x ^ n := by
simp only [rpow_def, ← Complex.ofReal_zpow, Complex.cpow_intCast, Complex.ofReal_intCast,
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 49 | 53 |
/-
Copyright (c) 2020 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Kevin Lacker
-/
import Mathlib.Tactic.Ring
/-!
# Identities
This file contains some "named" commutative ring identities.
-/
variable {R : Type*} [CommRing R] ... | ring
/-- Brahmagupta's identity, see <https://en.wikipedia.org/wiki/Brahmagupta%27s_identity>
| Mathlib/Algebra/Ring/Identities.lean | 24 | 26 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | add_zero := add_zero
add_assoc := by transfer
| Mathlib/Data/Num/Lemmas.lean | 342 | 343 |
/-
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.BigOperators.Group.Finset.Defs
/-! # Lemmas about coprimality with big products.
These lemmas ... | Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
| Mathlib/Data/Nat/GCD/BigOperators.lean | 24 | 26 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | theorem coe_image_subset_range : ↑(s.image f) ⊆ Set.range f :=
calc
↑(s.image f) = f '' ↑s := coe_image
_ ⊆ Set.range f := Set.image_subset_range f ↑s
| Mathlib/Data/Finset/Image.lean | 392 | 395 |
/-
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 ... | lemma _root_.LinearIsometryEquiv.adjoint_eq_symm (e : H ≃ₗᵢ[𝕜] K) :
adjoint (e : H →L[𝕜] K) = e.symm :=
let e' := (e : H →L[𝕜] K)
calc
adjoint e' = adjoint e' ∘L (e' ∘L e.symm) := by
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 476 | 480 |
/-
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.eq_of_mul_eq_zero {m n : ℕ} (h : m.Coprime n) (hmn : m * n = 0) :
| Mathlib/Data/Nat/GCD/Basic.lean | 212 | 213 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | section Order
variable {G'' G' : Type*}
[NormedAddCommGroup G'] [PartialOrder G'] [IsOrderedAddMonoid G'] [NormedSpace ℝ G']
[NormedAddCommGroup G''] [PartialOrder G''] [IsOrderedAddMonoid G''] [NormedSpace ℝ G'']
{T : Set α → G'' →L[ℝ] G'}
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 223 | 228 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... |
end PartialOrder
| Mathlib/Order/Bounds/Basic.lean | 888 | 889 |
/-
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 ... | Mathlib/Order/UpperLower/Basic.lean | 915 | 916 | |
/-
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.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.AffineSpace.Pointwise
import Mathlib.LinearAlgebra.Basis.SMul
/-!
# Affine bases and bary... | /-- TODO: generalize to include `SMul (P ≃ᵃ[k] P) (AffineBasis ι k P)`, which acts on `P` with a
`VAdd` version of a `DistribMulAction`. -/
instance [SMul G G'] [IsScalarTower G G' V] : IsScalarTower G G' (AffineBasis ι k V) where
smul_assoc _g _g' _b := DFunLike.ext _ _ fun _ => smul_assoc _ _ _
| Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 297 | 301 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 307 | 309 | |
/-
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.Data.Finite.Sum
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Finiteness.Ideal
import Mathlib.RingTheory.Ideal.Quotient.Operations
imp... |
/-- Induction principle for finitely presented ring homomorphisms.
For a property to hold for all finitely presented ring homs, it suffices for it to hold for
| Mathlib/RingTheory/FinitePresentation.lean | 451 | 454 |
/-
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.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | refine Or.inl ⟨l.mem_cons_self, ?_⟩
rw [prodExtendRight_apply_eq]
· refine Or.inr ⟨fun h => not_or_intro ha' not_mem_l ((List.mem_cons).mp h), ?_⟩
rw [prodExtendRight_apply_ne _ ha']
section congr
variable [DecidableEq β] [Fintype β]
@[simp]
theorem sign_prodExtendRight (a : α) (σ : Perm β) : sign (pro... | Mathlib/GroupTheory/Perm/Sign.lean | 526 | 545 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... | rw [uIoc_eq_union, aemeasurable_union_iff]
theorem aemeasurable_iff_measurable [μ.IsComplete] : AEMeasurable f μ ↔ Measurable f :=
⟨fun h => h.nullMeasurable.measurable_of_complete, fun h => h.aemeasurable⟩
theorem MeasurableEmbedding.aemeasurable_map_iff {g : β → γ} (hf : MeasurableEmbedding f) :
AEMeasurabl... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 232 | 239 |
/-
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.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | tendsto_iff_comap.2 (comap_dist_right_atTop c).ge
theorem tendsto_dist_left_cobounded_atTop (c : α) : Tendsto (dist c) (cobounded α) atTop :=
| Mathlib/Topology/MetricSpace/Bounded.lean | 137 | 139 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Convex.Deriv
/-!
# The function `x ↦ ... | not_continuousAt_of_tendsto tendsto_deriv_mul_log_nhdsWithin_zero nhdsWithin_le_nhds (by simp)
lemma deriv2_mul_log (x : ℝ) : deriv^[2] (fun x ↦ x * log x) x = x⁻¹ := by
| Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | 109 | 111 |
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