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) 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.Algebra.Polynomial.Cardinal
import Mathlib.RingTheory.Algebraic.Basic
/-!
### Cardinality of algebraic numbers
In this file, ... | section NonLift
variable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]
| Mathlib/Algebra/AlgebraicCard.lean | 93 | 95 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,514 | 1,515 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Nat.BitIndices
import Mathlib.Order.SupClosed
import Math... | classical
let s' : Finset α := {b ∈ s | b ∉ t ∧ a ≤ b}
have ⟨b, hb, hbmax⟩ := exists_maximal s' ⟨a, by simp [s', has, hat]⟩
simp only [s', mem_filter, and_imp] at hb hbmax
have ⟨c, hct, hcs, hbc⟩ := hst hb.1 hb.2.1
by_cases hcu : c ∈ u
· exact ⟨c, hcu, hcs, hb.2.2.trans hbc⟩
have ⟨d, hdu, hdt, hcd⟩ := h... | Mathlib/Combinatorics/Colex.lean | 105 | 117 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | forall_congr' fun _ ↦ differentiableAt_const_add_iff c
@[fun_prop]
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 300 | 302 |
/-
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 `... | induction n with
| zero => simpa using P.lie_h
| succ n ih =>
rw [pow_succ', Module.End.mul_apply, toEnd_apply_apply, Nat.cast_add, Nat.cast_one,
| Mathlib/Algebra/Lie/Sl2.lean | 112 | 115 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | (g'.lTensor _).comp (map f g) = map f (g'.comp g) := by
simp only [lTensor, rTensor, ← map_comp, id_comp, comp_id]
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,119 | 1,121 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Sites.Equ... | (terminalOpOfInitial initialIsInitial)).symm ≪≫ (PreservesTerminal.iso F)).hom using
injective_of_mono _
ext ⟨i⟩
exact i.elim
include hc hd hF hI in
/--
If `F` is a presheaf which `IsSheafFor` a presieve of arrows and the empty presieve, then it
preserves the product corresponding to the presieve o... | Mathlib/CategoryTheory/Sites/Preserves.lean | 129 | 145 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | theorem symm_image_eq (h : e.IsImage s t) : e.symm '' (e.target ∩ t) = e.source ∩ s :=
h.symm.image_eq
| Mathlib/Logic/Equiv/PartialEquiv.lean | 349 | 350 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... | let _ := fun i => Classical.decEq (G i)
refine Function.Injective.of_comp
(f := ((· • ·) : PushoutI φ → NormalWord d → NormalWord d)) ?_
intros _ _ h
exact eq_of_smul_eq_smul (fun w : NormalWord d =>
by simp_all [funext_iff, of_smul_eq_smul])
theorem base_injective (hφ : ∀ i, Function.Injective (φ i)) ... | Mathlib/GroupTheory/PushoutI.lean | 592 | 601 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | image_inter _ s t
theorem preimage_union (s t : Set β) : r.preimage (s ∪ t) = r.preimage s ∪ r.preimage t :=
| Mathlib/Data/Rel.lean | 210 | 212 |
/-
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | rcases UniformSpace.mem_nhds_iff.1 (IsOpen.mem_nhds hVo hxV) with ⟨U, hU, hUV⟩
rcases comp_symm_of_uniformity hU with ⟨U', hU', _, hUU'⟩
rcases h_basis.toHasBasis.mem_iff.1 hU' with ⟨k, -, hk⟩
rcases hsd.inter_open_nonempty (ball x <| t k) (isOpen_ball x (hto k))
⟨x, UniformSpace.mem_ball_self _... | Mathlib/Topology/UniformSpace/Cauchy.lean | 808 | 829 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1 :=
calc
| Mathlib/NumberTheory/Padics/Hensel.lean | 146 | 147 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 797 | 798 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... |
end CommMonoid
| Mathlib/Algebra/Group/Basic.lean | 208 | 209 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | rw [← sqrt_lt_sqrt_iff_of_pos (pow_pos hy _), sqrt_sq hy.le]
| Mathlib/Data/Real/Sqrt.lean | 208 | 209 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | cast nonnegg_comm (nonnegg_cases_right h)
section Norm
/-- The norm of an element of `ℤ[√d]`. -/
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 422 | 426 |
/-
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... |
theorem of_natCast_add (m : ℕ) (h : Irrational (m + x)) : Irrational x :=
| Mathlib/Data/Real/Irrational.lean | 250 | 251 |
/-
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.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered an... | constructor
· exact fun hx => Eq.mpr (congrArg (fun h ↦ h ≠ ⊤) (orderTop_of_ne hx)) WithTop.coe_ne_top
· contrapose!
simp_all only [orderTop_zero, implies_true]
| Mathlib/RingTheory/HahnSeries/Basic.lean | 252 | 256 |
/-
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.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,093 | 1,095 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | Nat.succ_add_sub_one, zero_add]
rw [degree_basis (Set.InjOn.mono hst hvt) hi, H, WithBot.coe_add, Nat.cast_withBot,
WithBot.add_lt_add_iff_right (@WithBot.coe_ne_bot _ (#s - 1))]
convert degree_interpolate_lt _
| Mathlib/LinearAlgebra/Lagrange.lean | 403 | 406 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | rcases h.left with ⟨a, H, ha⟩
use a, H
rwa [← h_top_iff_rpow_top a H]
· -- second, suppose both `(∑ i ∈ s, w i * z i) ^ p ≠ ⊤` and `∑ i ∈ s, (w i * z i ^ p) ≠ ⊤`,
-- and prove `((∑ i ∈ s, w i * z i) ^ p).toNNReal ≤ (∑ i ∈ s, (w i * z i ^ p)).toNNReal`,
-- by using `NNReal.rpow_arith_mean_le_arith_... | Mathlib/Analysis/MeanInequalitiesPow.lean | 233 | 276 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
/-!
# Projective seminorm on the tensor of a finite family of normed s... | 0 ≤ projectiveSeminormAux p := by
simp only [projectiveSeminormAux, Function.comp_apply]
refine List.sum_nonneg ?_
intro a
simp only [Multiset.map_coe, Multiset.mem_coe, List.mem_map, Prod.exists, forall_exists_index,
and_imp]
intro x m _ h
rw [← h]
exact mul_nonneg (norm_nonneg _) (Finset.prod_no... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 55 | 64 |
/-
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 | 167 | 174 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | choose! F hF using h
have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
set f := seqOfForallFinsetExistsAux P r h' with hf
| Mathlib/Data/Fintype/Basic.lean | 255 | 257 |
/-
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... | Mathlib/Data/Nat/PartENat.lean | 749 | 750 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
import Mathlib.CategoryTheory.Groupoid
/-!
# Facts about epimorphisms and monomorphisms.
The definitions of `Epi` an... | IsSplitEpi (f ≫ g) := IsSplitEpi.mk' <| hf.exists_splitEpi.some.comp hg.exists_splitEpi.some
| Mathlib/CategoryTheory/EpiMono.lean | 163 | 164 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | ⟨fun hf => ⟨hf.supermartingale, hf.submartingale⟩, fun ⟨hf₁, hf₂⟩ =>
⟨hf₁.1, fun i j hij => (hf₁.2.1 i j hij).antisymm (hf₂.2.1 i j hij)⟩⟩
| Mathlib/Probability/Martingale/Basic.lean | 128 | 129 |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... | (gc_set _).le_u_l s
theorem ideal_le_vanishingIdeal_zeroLocus (I : Ideal A) :
I ≤ (vanishingIdeal (zeroLocus 𝒜 I)).toIdeal :=
(gc_ideal _).le_u_l I
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 144 | 149 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | (total : fst ≫ T.mor₁ + T.mor₂ ≫ inr = 𝟙 T.obj₂) :
BinaryBiproductData T.obj₁ T.obj₃ := by
have : Mono T.mor₁ := T.mono₁ hT hT₀
have eq : fst ≫ T.mor₁ = 𝟙 T.obj₂ - T.mor₂ ≫ inr := by rw [← total, add_sub_cancel_right]
| Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 470 | 473 |
/-
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... | instance instAddZeroClass : AddZeroClass (α →₀ M) :=
fast_instance% DFunLike.coe_injective.addZeroClass _ coe_zero coe_add
| Mathlib/Data/Finsupp/Defs.lean | 491 | 493 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | · rw [isOfFinOrder_iff_pow_eq_one]
exact Exists.intro n (id ⟨hn, hx⟩)
| Mathlib/GroupTheory/OrderOfElement.lean | 308 | 309 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | · cases nonempty_fintype (K ≃ₐ[k] K)
rw [even_iff_two_dvd, ← not_isUnramified_iff_card_stabilizer_eq_two.mp hw]
exact Subgroup.card_subgroup_dvd_card (Stab w)
· convert Even.zero
by_contra e
exact H (Nat.finite_of_card_ne_zero e)
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 948 | 953 |
/-
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.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | | 0 => Int.sign_zero
| n + 1 =>
show (Nat.gcdA _ (n + 1) : ZMod (n + 1)) = 0 by
rw [val_zero]
unfold Nat.gcdA Nat.xgcd Nat.xgcdAux
| Mathlib/Data/ZMod/Basic.lean | 699 | 703 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Separation.Hausdorff
/-!
# Order-closed topologies
In this file we ... |
protected theorem SuccOrder.nhdsGT [SuccOrder α] : 𝓝[>] a = ⊥ := PredOrder.nhdsLT (α := αᵒᵈ)
| Mathlib/Topology/Order/OrderClosed.lean | 560 | 562 |
/-
Copyright (c) 2023 Claus Clausen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Claus Clausen, Patrick Massot
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.CDF
import Mathlib.Probability.Distributions.Gamma
/-! # Exponential distributions over ... | lemma exponentialPDF_eq (r x : ℝ) :
exponentialPDF r x = ENNReal.ofReal (if 0 ≤ x then r * exp (-(r * x)) else 0) := by
rw [exponentialPDF, exponentialPDFReal, gammaPDFReal]
simp only [rpow_one, Gamma_one, div_one, sub_self, rpow_zero, mul_one]
| Mathlib/Probability/Distributions/Exponential.lean | 47 | 50 |
/-
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.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... |
variable [AddCommMonoid α] [Mul α]
theorem vec2_dotProduct' {a₀ a₁ b₀ b₁ : α} : ![a₀, a₁] ⬝ᵥ ![b₀, b₁] = a₀ * b₀ + a₁ * b₁ := by
| Mathlib/Data/Matrix/Notation.lean | 504 | 507 |
/-
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.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | end
namespace Primcodable
variable {α : Type*} {β : Type*}
variable [Primcodable α] [Primcodable β]
| Mathlib/Computability/Primrec.lean | 782 | 787 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | /-- The closed ball of radius `δ` centered at a point of `E` is included in the closed
thickening of `E`. -/
theorem closedBall_subset_cthickening {α : Type*} [PseudoMetricSpace α] {x : α} {E : Set α}
(hx : x ∈ E) (δ : ℝ) : closedBall x δ ⊆ cthickening δ E := by
refine (closedBall_subset_cthickening_singleton _ _... | Mathlib/Topology/MetricSpace/Thickening.lean | 517 | 525 |
/-
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 | 1,742 | 1,765 | |
/-
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... | rfl
@[simp]
theorem sumInv_apply (x : AdicCompletion I (⨁ j, M j)) (j : ι) :
(sumInv I M x) j = map I (component R ι _ j) x := by
| Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 279 | 283 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 551 | 552 | |
/-
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.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,172 | 1,173 | |
/-
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
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 292 | 292 | |
/-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Coset.Basic
/-!
# Double cosets
This file defines double cosets for two subgroups `H K` o... | end Doset
| Mathlib/GroupTheory/DoubleCoset.lean | 186 | 196 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
/-!
# Functors with two arguments
This file defines bifunctors.
A bifunctor is a function `F : Type* → Type* ... | theorem comp_snd {α β₀ β₁ β₂} (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α β₀) :
snd g' (snd g x) = snd (g' ∘ g) x := by simp [snd, bimap_bimap]
| Mathlib/Control/Bifunctor.lean | 86 | 87 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.CompHausLike.Limits
import Mathlib.Topology.Category.LightProfinite.Basic
/-!
# Explicit limits and colimits
This file applies ... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 220 | 224 | |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | · apply hζ.pow_ne_one_of_pos_of_lt zero_lt_one (one_lt_pow₀ hp.out.one_lt (by simp))
rw [sub_eq_zero] at h
simpa using congrArg (algebraMap _ K) h
rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,
| Mathlib/NumberTheory/Cyclotomic/Rat.lean | 295 | 298 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | rw [← map_nonsingular _ _ hf, AlgHom.toRingHom_eq_coe, map_baseChange, RingHom.coe_coe]
lemma baseChange_negPolynomial : (W'.baseChange B).toAffine.negPolynomial =
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 855 | 857 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | theorem y_zpow_pos {a : Solution₁ d} (hax : 0 < a.x) (hay : 0 < a.y) {n : ℤ} (hn : 0 < n) :
0 < (a ^ n).y := by
lift n to ℕ using hn.le
norm_cast at hn ⊢
rw [← Nat.succ_pred_eq_of_pos hn]
exact y_pow_succ_pos hax hay _
/-- If `(x, y)` is a solution with `x` positive, then all its powers have positive `x`. ... | Mathlib/NumberTheory/Pell.lean | 264 | 277 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | posFittingCompOf_le_lowerCentralSeries, forall_const]
variable (L)
| Mathlib/Algebra/Lie/Weights/Basic.lean | 443 | 445 |
/-
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 | 1,627 | 1,631 | |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | (hf.hasFDerivAt.hasLineDerivAt v).lineDeriv
theorem LineDifferentiableWithinAt.mono_of_mem_nhdsWithin (h : LineDifferentiableWithinAt 𝕜 f s x v)
(hst : s ∈ 𝓝[t] x) : LineDifferentiableWithinAt 𝕜 f t x v :=
(h.hasLineDerivWithinAt.mono_of_mem_nhdsWithin hst).lineDifferentiableWithinAt
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 275 | 279 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | inj' := map.injective f.injective
lemma lift_comp_map {g : γ → α} (f : {f : α → α → β // ∀ a₁ a₂, f a₁ a₂ = f a₂ a₁}) :
| Mathlib/Data/Sym/Sym2.lean | 270 | 272 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | theorem nontrivial_of_einfsep_lt_top (hs : s.einfsep < ∞) : s.Nontrivial := by
rcases einfsep_lt_top.1 hs with ⟨_, hx, _, hy, hxy, _⟩
exact ⟨_, hx, _, hy, hxy⟩
| Mathlib/Topology/MetricSpace/Infsep.lean | 74 | 76 |
/-
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 | 958 | 960 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | · rw [le_div_iff_mul_le h_spand, mul_assoc, mul_left_comm, one_div_spanSingleton,
spanSingleton_mul_spanSingleton, inv_mul_cancel₀ hd, spanSingleton_one, mul_one]
theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 737 | 755 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 1,128 | 1,130 | |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | Mathlib/MeasureTheory/Constructions/Pi.lean | 909 | 915 | |
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | Subtype.ext_iff
| Mathlib/Data/Rat/Lemmas.lean | 316 | 317 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... |
/-- When lowered to a prop, `Matrix.diagonalInvertibleEquivInvertible` forms an `iff`. -/
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 551 | 552 |
/-
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | simpa only [Function.comp_def, f.apply_symm_apply] using H.comp_injective f.symm.injective
theorem CauchySeq.subseq_subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
(hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n,... | Mathlib/Topology/UniformSpace/Cauchy.lean | 211 | 215 |
/-
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.Algebra.Defs
import Mathlib.Algebra.Order.Module.Defs
/-!
# Ordered algebras
This file proves properties of algebras where both rings are ordered... | @[mono] lemma algebraMap_mono : Monotone (algebraMap α β) :=
fun a₁ a₂ ha ↦ by
simpa only [Algebra.algebraMap_eq_smul_one] using smul_le_smul_of_nonneg_right ha zero_le_one
| Mathlib/Algebra/Order/Module/Algebra.lean | 25 | 27 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | def mkCache {α : Q(Type u)} (sα : Q(CommSemiring $α)) : MetaM (Cache sα) :=
return {
| Mathlib/Tactic/Ring/Basic.lean | 900 | 901 |
/-
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.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
/-!
# The category of schemes
A schem... |
/-- An isomorphism of schemes induces a homeomorphism of the underlying topological spaces. -/
noncomputable def homeoOfIso {X Y : Scheme.{u}} (e : X ≅ Y) : X ≃ₜ Y :=
TopCat.homeoOfIso (forgetToTop.mapIso e)
@[simp]
| Mathlib/AlgebraicGeometry/Scheme.lean | 219 | 224 |
/-
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.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | theorem val_add {n : ℕ} [NeZero n] (a b : ZMod n) : (a + b).val = (a.val + b.val) % n := by
cases n
· cases NeZero.ne 0 rfl
· apply Fin.val_add
| Mathlib/Data/ZMod/Basic.lean | 626 | 629 |
/-
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
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# One-dimensional derivatives
This ... | HasDerivAtFilter.congr_of_eventuallyEq h h₁ hx
theorem Filter.EventuallyEq.hasDerivWithinAt_iff (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
HasDerivWithinAt f₁ f' s x ↔ HasDerivWithinAt f f' s x :=
⟨fun h' ↦ h'.congr_of_eventuallyEq h₁.symm hx.symm, fun h' ↦ h'.congr_of_eventuallyEq h₁ hx⟩
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 546 | 551 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,620 | 1,623 | |
/-
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... | rw [ncard, h.encard_eq_coe_toFinset_card, Nat.card_eq_fintype_card,
toFinite_toFinset, toFinset_card, ENat.toNat_coe]
have := infinite_coe_iff.2 h
rw [ncard, h.encard_eq, Nat.card_eq_zero_of_infinite, ENat.toNat_top]
| Mathlib/Data/Set/Card.lean | 510 | 513 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
/-!
# Derivations
This file defines derivation. A derivat... | theorem zero_apply (a : A) : (0 : Derivation R A M) a = 0 :=
rfl
| Mathlib/RingTheory/Derivation/Basic.lean | 177 | 179 |
/-
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
/-... | [IsLocallyInjective J f₂] [IsLocallySurjective J f₂] :
IsLocallySurjective J (f₁ ≫ f₂) ↔ IsLocallySurjective J f₁ := by
constructor
· intro
exact isLocallySurjective_of_isLocallySurjective_of_isLocallyInjective J f₁ f₂
· intro
infer_instance
attribute [local instance] Types.instFunLike Types.inst... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 253 | 261 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Constructions
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
import Mathlib.Topology.Algebra.Monoid.Defs
/-!
# To... | theorem continuous_eraseIdx {n : ℕ} {i : Fin (n + 1)} :
Continuous (Vector.eraseIdx i : Vector α (n + 1) → Vector α n) :=
continuous_iff_continuousAt.mpr fun ⟨_a, _l⟩ => continuousAt_eraseIdx
end List.Vector
| Mathlib/Topology/List.lean | 206 | 212 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 663 | 663 | |
/-
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 le_log_of_pow_le (hx : 0 < x) (h : x ^ n ≤ y) : n * log x ≤ log y :=
le_log_of_rpow_le hx (rpow_natCast _ _ ▸ h)
lemma le_log_of_zpow_le {n : ℤ} (hx : 0 < x) (h : x ^ n ≤ y) : n * log x ≤ log y :=
le_log_of_rpow_le hx (rpow_intCast _ _ ▸ h)
lemma rpow_le_of_le_log (hy : 0 < y) (h : log x ≤ z * log y) : x ≤ ... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 821 | 827 |
/-
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 | 1,136 | 1,140 | |
/-
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.Analysis.Asymptotics.Lemmas
import Mathlib.Analysis.Normed.Module.Basic
/-!
# Asymptotic equivalence up to a constant
In this file we define `Asymp... |
lemma IsTheta.isTheta_congr_right (h : f' =Θ[l] g') : k =Θ[l] f' ↔ k =Θ[l] g' :=
h.isBigO_congr_right.and h.isBigO_congr_left
| Mathlib/Analysis/Asymptotics/Theta.lean | 166 | 168 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 612 | 620 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
/-!
# The M construction as a multivariate polyn... | refine
PFunctor.M.bisim (fun a₁ a₂ => ∃ x y, R x y ∧ x.1 = a₁ ∧ y.1 = a₂) ?_ _ _
⟨⟨a₁, f₁⟩, ⟨a₂, f₂⟩, r, rfl, rfl⟩
rintro _ _ ⟨⟨a₁, f₁⟩, ⟨a₂, f₂⟩, r, rfl, rfl⟩
rcases h _ _ r with ⟨a', f', f₁', f₂', e₁, e₂, h'⟩
rcases M.bisim_lemma P e₁ with ⟨g₁', e₁', rfl, rfl⟩
rcases M.bisim_lemma P ... | Mathlib/Data/PFunctor/Multivariate/M.lean | 223 | 233 |
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