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) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | hg.mono hf <| h.mono fun _x hx => le_trans hx (le_abs_self _)
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 557 | 558 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Order.Field
impo... | rw [sq, ← Nat.cast_mul, ← Nat.cast_mul, Nat.cast_le]
exact Nat.mul_le_mul (hP.card_part_le_average_add_one hU)
(hP.card_part_le_average_add_one hV)
· rw [mul_right_comm _ ε, mul_comm ε]
apply mul_lt_mul_of_pos_right _ hε
norm_cast
exact aux (P.parts_nonempty hA.ne_empty).card_pos
lemma IsEq... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 359 | 373 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.Unbundled
/-!
# Lexicographic order on a sigma type
This defines the lexicographical order of two arbitrary... | h.mono hr fun _ _ _ => id
theorem Lex.mono_right {r : ι → ι → Prop} {s₁ s₂ : ∀ i, α i → α i → Prop}
(hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ' i, α i} (h : Lex r s₁ a b) : Lex r s₂ a b :=
h.mono (fun _ _ => id) hs
| Mathlib/Data/Sigma/Lex.lean | 170 | 175 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,867 | 1,874 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
/-!
# Quasi-compact morphisms
A morp... | refine (Set.inter_eq_right.mpr
(SetLike.coe_subset_coe.2 <| RingedSpace.basicOpen_le _ _)).symm.trans ?_
rw [e, Set.iUnion₂_inter]
| Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 104 | 106 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | Cochain.id_comp]
@[simp]
lemma inl_snd_assoc {K : CochainComplex C ℤ} {d e f : ℤ} (γ : Cochain G K d)
(he : 0 + d = e) (hf : -1 + e = f) :
(inl φ).comp ((snd φ).comp γ he) hf = 0 := by
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 127 | 132 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | def scMapIso {S₁ S₂ : ComposableArrows C n} (e : S₁ ≅ S₂)
(h₁ : S₁.IsComplex) (h₂ : S₂.IsComplex)
(i : ℕ) (hi : i + 2 ≤ n := by omega) :
S₁.sc h₁ i ≅ S₂.sc h₂ i where
hom := scMap e.hom h₁ h₂ i
inv := scMap e.inv h₂ h₁ i
hom_inv_id := by ext <;> dsimp <;> simp
inv_hom_id := by ext <;> dsimp <;> simp... | Mathlib/Algebra/Homology/ExactSequence.lean | 168 | 176 |
/-
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 | 636 | 637 | |
/-
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... | rw [nhds_eq_comap_uniformity, tendsto_comap_iff]; rfl
theorem tendsto_nhds_left {f : Filter β} {u : β → α} {a : α} :
Tendsto u f (𝓝 a) ↔ Tendsto (fun x => (u x, a)) f (𝓤 α) := by
rw [nhds_eq_comap_uniformity', tendsto_comap_iff]; rfl
| Mathlib/Topology/UniformSpace/Basic.lean | 917 | 922 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | Mathlib/Data/Finmap.lean | 619 | 622 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... |
/-- Construct an isomorphism of chain complexes from isomorphism of the objects
which commute with the differentials. -/
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 496 | 498 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Inv
/-!
# `norm_num` extension for equalities
-/
variable {α : Type*}
open Lean Meta Qq
namespace Mathlib.Meta.NormNum
theorem isNa... | theorem isRat_eq_false [Ring α] [CharZero α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) = Int.mul nb (.ofNat da)) = false → ¬a = b
| _, _, _, _, _, _, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
rw [Rat.invOf_denom_swap]; exact mod_cast of_decide_eq_false h
| Mathlib/Tactic/NormNum/Eq.lean | 31 | 35 |
/-
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.OuterMeasure.Induced
import Mathlib.MeasureTheory.OuterMeasure.AE
import Mathlib.Order.Filter.CountableInter
/-!
# Measu... | (s : Set α) (hs : MeasurableSet s) : ofMeasurable m m0 mU s = m s hs :=
inducedOuterMeasure_eq m0 mU hs
| Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 136 | 138 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | @[simp]
theorem rotate_eq_nil_iff {l : List α} {n : ℕ} : l.rotate n = [] ↔ l = [] := by
| Mathlib/Data/List/Rotate.lean | 162 | 163 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.LocalizerMorphism
/-!
# Resolutions for a morphism of localizers
Given a morphism of localizers `Φ : LocalizerMorphism W₁ W₂` (i.e.... | lemma essSurj_of_hasRightResolutions [Φ.HasRightResolutions] : (Φ.functor ⋙ L₂).EssSurj where
mem_essImage X₂ := by
have := Localization.essSurj L₂ W₂
have R : Φ.RightResolution (L₂.objPreimage X₂) := Classical.arbitrary _
exact ⟨R.X₁, ⟨(Localization.isoOfHom L₂ W₂ _ R.hw).symm ≪≫ L₂.objObjPreimageIso X₂⟩... | Mathlib/CategoryTheory/Localization/Resolution.lean | 284 | 294 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | (H : ∀ a (H : a < o), P <| toLeftMovesNim ⟨a, H⟩) : P i := by
| Mathlib/SetTheory/Game/Nim.lean | 111 | 111 |
/-
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 _root_.Finset.infsep_pos_iff_nontrivial (s : Finset α) :
0 < (s : Set α).infsep ↔ (s : Set α).Nontrivial :=
infsep_pos_iff_nontrivial_of_finite
end MetricSpace
end Infsep
| Mathlib/Topology/MetricSpace/Infsep.lean | 473 | 481 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... |
variable {α β : Type*} [NormedField β] {u v : α → β} {l : Filter α}
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 172 | 173 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Homology
/-!
# Quasi-isomorphisms of short complexes
This file introduces the typeclass `QuasiIso φ` for a morphism `φ : S₁ ⟶ S₂... | lemma quasiIso_iff_isIso_liftCycles (φ : S₁ ⟶ S₂)
(hf₁ : S₁.f = 0) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) :
QuasiIso φ ↔ IsIso (S₂.liftCycles φ.τ₂ (by rw [φ.comm₂₃, hg₁, zero_comp])) := by
let H : LeftHomologyMapData φ (LeftHomologyData.ofZeros S₁ hf₁ hg₁)
(LeftHomologyData.ofIsLimitKernelFork S₂ hf₂ _ S₂.cy... | Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean | 167 | 174 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... |
/-- The product of two `s`-factored numbers is again `s`-factored. -/
lemma mul_mem_factoredNumbers {s : Finset ℕ} {m n : ℕ} (hm : m ∈ factoredNumbers s)
(hn : n ∈ factoredNumbers s) :
m * n ∈ factoredNumbers s := by
have hm' := primeFactors_subset_of_mem_factoredNumbers hm
have hn' := primeFactors_subset_... | Mathlib/NumberTheory/SmoothNumbers.lean | 138 | 145 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finite.Basic
/-!
# Lagrange's four square theorem
The main result in this file is `sum_four_squares`,
a proof that every natural number is th... | have : (∑ x, f (σ x) ^ 2) = ∑ x, f x ^ 2 := Equiv.sum_comp σ (f · ^ 2)
simpa only [← hx, ← hy, Fin.sum_univ_four, add_assoc] using this
/-- Lagrange's **four squares theorem** for a prime number. Use `Nat.sum_four_squares` instead. -/
protected theorem Prime.sum_four_squares {p : ℕ} (hp : p.Prime) :
∃ a b c d ... | Mathlib/NumberTheory/SumFourSquares.lean | 105 | 128 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Identically distributed random variables
Two random variable... | strongly measurable. So is the second function, but use `h.symm.aestronglyMeasurable_fst` as
`h.aestronglyMeasurable_snd` has a different meaning. -/
theorem aestronglyMeasurable_fst [TopologicalSpace γ] [MetrizableSpace γ] [OpensMeasurableSpace γ]
[SecondCountableTopology γ] (h : IdentDistrib f g μ ν) : AEStrongly... | Mathlib/Probability/IdentDistrib.lean | 141 | 145 |
/-
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 | 1,556 | 1,559 | |
/-
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
import Mathlib.Algebra.Order.ZeroLEOne
import Mathli... |
lemma zero_le_three [Preorder α] [ZeroLEOneClass α] [AddLeftMono α] :
(0 : α) ≤ 3 := by
rw [← two_add_one_eq_three]
| Mathlib/Algebra/Order/Monoid/NatCast.lean | 32 | 35 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
import Mathlib.Algebra.Ring.Action.Rat
import Mathlib.Tactic.FastInstance
/-!
# Cauchy completion
This fi... | have h₃ : f * inv f hf * g = (f * inv f hf) * g := by simp [mul_assoc]
have h₄ : g - f * inv f hf * g = (1 - f * inv f hf) * g := by rw [h₂, h₃, ← sub_mul]
have h₅ : g - f * inv f hf * g = g * (1 - f * inv f hf) := by rw [h₄, mul_comm]
have h₆ : g - f * inv f hf * g = g * (1 - in... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 371 | 383 |
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | @[simp] lemma in₁₀_iff : (graph t).Adj (in₁ b) (in₀ a) ↔ ∃ c, (a, b, c) ∈ t :=
⟨by rintro ⟨⟩; exact ⟨_, ‹_›⟩, fun ⟨_, h⟩ ↦ in₁₀ h⟩
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 74 | 75 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,115 | 1,116 | |
/-
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.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... | else by
push_neg at hif
rw [← Order.succ_le_iff, ← WithBot.coe_zero, WithBot.orderSucc_coe, Nat.succ_eq_succ] at hif
exact splits_of_map_degree_eq_one i ((degree_map_le.trans hf).antisymm hif)
theorem splits_of_degree_eq_one {f : K[X]} (hf : degree f = 1) : Splits i f :=
| Mathlib/Algebra/Polynomial/Splits.lean | 77 | 82 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `norm_num` tactic so that it can be... | (hr : jacobiSymNat a c = r) : jacobiSymNat a b = r := by
have ha' : legendreSym 2 a = 1 := by
simp only [legendreSym.mod 2 a, Int.ofNat_mod_ofNat, ha]
decide
rcases eq_or_ne c 0 with (rfl | hc')
· rw [← hr, Nat.eq_zero_of_dvd_of_div_eq_zero (Nat.dvd_of_mod_eq_zero hb) hc]
· haveI : NeZero c := ⟨hc'⟩... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 109 | 117 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... |
theorem chainHeight_union_le : (s ∪ t).chainHeight ≤ s.chainHeight + t.chainHeight := by
classical
refine iSup₂_le fun l hl ↦ ?_
let l₁ := l.filter (· ∈ s)
let l₂ := l.filter (· ∈ t)
have hl₁ : ↑l₁.length ≤ s.chainHeight := by
apply Set.length_le_chainHeight_of_mem_subchain
exact ⟨hl.1.su... | Mathlib/Order/Height.lean | 281 | 306 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | rw [smul_ball_zero (norm_pos_iff.1 <| hr₀.trans_le ha), p.mem_ball_zero]
rw [p.mem_ball_zero] at hx
exact hx.trans (hr.trans_le <| by gcongr)
/-- Seminorm-balls at the origin are absorbent. -/
protected theorem absorbent_ball_zero (hr : 0 < r) : Absorbent 𝕜 (ball p (0 : E) r) :=
absorbent_iff_forall_absorbs_s... | Mathlib/Analysis/Seminorm.lean | 930 | 937 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | (prod_le_iff.2 ⟨
map_le_iff_le_comap.2 <| closure_le _ |>.2 fun _x hx => subset_closure ⟨hx, ht⟩,
map_le_iff_le_comap.2 <| closure_le _ |>.2 fun _y hy => subset_closure ⟨hs, hy⟩⟩)
/-- Product of subgroups is isomorphic to their product as groups. -/
@[to_additive prodEquiv
"Product of additive su... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 150 | 156 |
/-
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.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | · exact fun e he ↦ (hIB' he).elim (fun h' ↦ (h' (hI he)).elim) id
rw [subset_diff, and_iff_right hB'.subset_ground, disjoint_comm]
| Mathlib/Data/Matroid/Basic.lean | 512 | 513 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Chris Hughes, Mantas Bakšys
-/
import Mathlib.Data.List.Basic
import Mathlib.Order.BoundedOrder.Lattice
import Mathlib.Data.List.Induction
import Mathlib.Order.MinMax
import Ma... | · simpa [h y mem_cons_self] using IH fun x hx => h x <| mem_cons_of_mem _ hx
theorem le_max_of_le {l : List α} {a x : α} (hx : x ∈ l) (h : a ≤ x) : a ≤ l.foldr max ⊥ := by
induction' l with y l IH
· exact absurd hx not_mem_nil
· obtain hl | hl := hx
· simp only [foldr, foldr_cons]
| Mathlib/Data/List/MinMax.lean | 499 | 505 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | private lemma exists_mul_right_lt₀ {a b c : α} (hc : a * b < c) : ∃ b' > b, a * b' < c := by
simp_rw [mul_comm a] at hc ⊢; exact exists_mul_left_lt₀ hc
| Mathlib/Algebra/Order/Field/Basic.lean | 616 | 617 |
/-
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.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on mul... | · intros
simp [*]
theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
| Mathlib/Algebra/MvPolynomial/Rename.lean | 206 | 210 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | ‖y‖ / Real.tan (angle x (x + y)) = ‖x‖ := by
rw [tan_angle_add_of_inner_eq_zero h]
rcases h0 with (h0 | h0)
· simp [h0]
· rw [div_div_eq_mul_div, mul_comm, div_eq_mul_inv, mul_inv_cancel_right₀ (norm_ne_zero_iff.2 h0)]
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 197 | 202 |
/-
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.... | φ.Realize (v xs) (xs ∘ Fin.natAdd _) := by
induction φ with
| Mathlib/ModelTheory/Semantics.lean | 349 | 350 |
/-
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.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower.Basic
/-!
# Tensor algebras as direct sums of tensor powers
In this file we show... | -- It seems the side condition `h` is not applied by `simpNF`.
@[simp, nolint simpNF]
theorem mk_reindex_cast {n m : ℕ} (h : n = m) (x : ⨂[R]^n M) :
GradedMonoid.mk (A := fun i => (⨂[R]^i) M) m
| Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 107 | 110 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Fintype.Order
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.Quotients
import... |
end DirectLimit
variable (G)
namespace DirectLimit
| Mathlib/ModelTheory/DirectLimit.lean | 113 | 118 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Defs
import Batteries.Data.List.Basic
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two... |
theorem suffixLevenshtein_eq_tails_map (xs ys) :
(suffixLevenshtein C xs ys).1 = xs.tails.map fun xs' => levenshtein C xs' ys := by
induction xs with
| nil =>
simp only [suffixLevenshtein_nil', List.tails, List.map_cons, List.map]
| cons x xs ih =>
| Mathlib/Data/List/EditDistance/Defs.lean | 265 | 271 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | ext k j
by_cases h : k = e j
· subst h
simp
· simp only [ι_desc_assoc, Category.assoc, ne_eq, lift_π]
rw [biproduct.ι_π_ne, biproduct.ι_π_ne_assoc]
| Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 640 | 645 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | · rw [degree_eq_natDegree hp]
exact le_degree_of_ne_zero h
theorem natDegree_le_iff_degree_le {n : ℕ} : natDegree p ≤ n ↔ degree p ≤ n :=
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 132 | 135 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Tactic.Basic
import Mathlib.Order.Filter.Basic
/-!
# The `peel` tactic
`peel h with h' idents*` tries to apply `forall_imp` (or `Exists.imp`, or `Filte... | private theorem eventually_congr {α : Type*} {p q : α → Prop} {f : Filter α}
(hq : ∀ (x : α), p x ↔ q x) : (∀ᶠ (x : α) in f, p x) ↔ ∀ᶠ (x : α) in f, q x := by
congr! 2; exact hq _
| Mathlib/Tactic/Peel.lean | 97 | 99 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | variable [NoZeroDivisors R]
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 337 | 338 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | @[simp]
| Mathlib/Order/Heyting/Basic.lean | 401 | 401 |
/-
Copyright (c) 2015 Leonardo de Moura. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.List.Basic
import Mathlib.Data.Prod.Basic
/-!
# Lists in product and sigma types
This file proves basic properties of `Lis... | theorem length_product (l₁ : List α) (l₂ : List β) :
length (l₁ ×ˢ l₂) = length l₁ * length l₂ := by
induction' l₁ with x l₁ IH
· exact (Nat.zero_mul _).symm
| Mathlib/Data/List/ProdSigma.lean | 45 | 48 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... |
theorem constr_pow_mul (pb : PowerBasis A S) {y : S'} (hy : aeval y (minpoly A pb.gen) = 0)
(x x' : S) : pb.basis.constr A (fun i => y ^ (i : ℕ)) (x * x') =
| Mathlib/RingTheory/PowerBasis.lean | 266 | 268 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.sin_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 100 | 104 |
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... |
/-- A function is continuous if and only if its composition with a partial homeomorphism
on the left is continuous and its image is contained in the source. -/
theorem continuous_iff_continuous_comp_left {f : Z → X} (h : f ⁻¹' e.source = univ) :
Continuous f ↔ Continuous (e ∘ f) := by
simp only [continuous_iff_c... | Mathlib/Topology/PartialHomeomorph.lean | 1,072 | 1,081 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | lemma even_sub' (h : n ≤ m) : Even (m - n) ↔ (Odd m ↔ Odd n) := by
rw [even_sub h, ← not_odd_iff_even, ← not_odd_iff_even, not_iff_not]
| Mathlib/Algebra/Ring/Parity.lean | 247 | 248 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... | theorem eq_of_uniformity_basis {α : Type*} [UniformSpace α] [T0Space α] {ι : Sort*}
{p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
(h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
| Mathlib/Topology/UniformSpace/Separation.lean | 155 | 157 |
/-
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.Algebra.Order.Group.Nat
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Logic.Enco... | refine le_antisymm (Nat.find_le ?_) (zero_le _)
| Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 221 | 221 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | lemma abs_le_of_sq_le_sq (h : a ^ 2 ≤ b ^ 2) (hb : 0 ≤ b) : |a| ≤ b := by
rwa [← abs_of_nonneg hb, ← sq_le_sq]
| Mathlib/Algebra/Order/Ring/Abs.lean | 116 | 118 |
/-
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... | with `y`, then `x * y` has the same order as `y`. -/
@[to_additive addOrderOf_add_eq_right_of_forall_prime_mul_dvd
"If each prime factor of
`addOrderOf x` has higher multiplicity in `addOrderOf y`, and `x` commutes with `y`,
then `x + y` has the same order as `y`."]
| Mathlib/GroupTheory/OrderOfElement.lean | 436 | 440 |
/-
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.CharP.Algebra
import Mathlib.Data.Multiset.Fintype
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
/... | AlgHom.coe_toRingHom, MvPolynomial.aeval_X, dif_pos h,
← (finEquivRoots (Monics.splits_finsetProd h)).symm.prod_comp, Equiv.apply_symm_apply]
rw [Finset.prod_coe_sort (f := fun x : _ × ℕ ↦ X - C x.1), (Multiset.toEnumFinset _)
|>.prod_eq_multiset_prod, ← Function.comp_def (X - C ·) Prod.fst, ← Multiset.ma... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 85 | 94 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... | apply LinearEquiv.toLinearMap_injective
ext f
simp only [LinearEquiv.trans_apply, LinearEquiv.coe_coe, reindex_tprod,
LinearMap.coe_compMultilinearMap, Function.comp_apply, MultilinearMap.domDomCongr_apply,
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 734 | 737 |
/-
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.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... |
/-- If the cardinality is positive, that means it is a finite type, so there is
| Mathlib/SetTheory/Cardinal/Finite.lean | 173 | 174 |
/-
Copyright (c) 2020 Bhavik Mehta, Edward Ayers, Thomas Read. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers, Thomas Read
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.Cate... | /-- The internal element which points at the given morphism. -/
def internalizeHom (f : A ⟶ Y) : 𝟙_ C ⟶ A ⟹ Y :=
| Mathlib/CategoryTheory/Closed/Cartesian.lean | 239 | 240 |
/-
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.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | theorem symm_apply (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
e.symm b y = cast (congr_arg E (e.symm_coe_proj hb)) (e.toPartialHomeomorph.symm (b, y)).2 :=
dif_pos hb
| Mathlib/Topology/FiberBundle/Trivialization.lean | 538 | 541 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Eric Wieser
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.Star
import Mathlib.LinearAlgebra.... |
theorem HasSum.matrix_diagonal [DecidableEq n] {f : X → n → R} {a : n → R} (hf : HasSum f a) :
HasSum (fun x => diagonal (f x)) (diagonal a) :=
hf.map (diagonalAddMonoidHom n R) continuous_id.matrix_diagonal
| Mathlib/Topology/Instances/Matrix.lean | 310 | 314 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... |
/-- If a function admits a power series expansion within a set on a ball, then it is
continuous there. -/
protected theorem HasFPowerSeriesWithinOnBall.continuousOn
(hf : HasFPowerSeriesWithinOnBall f p s x r) :
ContinuousOn f (insert x s ∩ EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
| Mathlib/Analysis/Analytic/Basic.lean | 1,268 | 1,274 |
/-
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... | theorem mul_exp_pos {a₁ a₂ : ℕ} (n) (h₁ : 0 < a₁) (h₂ : 0 < a₂) : 0 < a₁ ^ n * a₂ :=
Nat.mul_pos (Nat.pow_pos h₁) h₂
| Mathlib/Tactic/Ring/Basic.lean | 665 | 666 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | eq_top_iff.2 fun x _ =>
FreeMonoid.recOn x (one_mem _) fun _x _xs hxs =>
mul_mem (subset_closure <| Set.mem_range_self _) hxs
end FreeMonoid
| Mathlib/Algebra/Group/Submonoid/Membership.lean | 151 | 155 |
/-
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.Module.Equiv.Defs
import Mathlib.Data.DFinsupp.Module
import Mathlib.Data.Finsupp.SMul
/-!
# Conversion between `Finsupp` and homogeneous `DFinsupp`... | @[simp]
theorem DFinsupp.toFinsupp_toDFinsupp (f : Π₀ _ : ι, M) : f.toFinsupp.toDFinsupp = f :=
DFunLike.coe_injective rfl
| Mathlib/Data/Finsupp/ToDFinsupp.lean | 123 | 126 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,377 | 1,378 | |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Matrix.CharP
/-!
# Results o... | Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean | 61 | 62 | |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
/-! # Results about inverses in Clifford algebras
This contains some basic results about the inversion of vectors... | invOf_mul_self, one_smul]
/-- $a^{-1}ba$ is a vector. -/
theorem invOf_ι_mul_ι_mul_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
| Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 44 | 47 |
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Logic.Lemmas
import Batteries.Data.List.Basic
/-!
# Paths in quivers
Given a quiver `V`, we def... | theorem mapPath_toPath {a b : V} (f : a ⟶ b) : F.mapPath f.toPath = (F.map f).toPath :=
rfl
@[simp]
| Mathlib/Combinatorics/Quiver/Path.lean | 227 | 230 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 625 | 631 | |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | exact ⟨q ⊗ₜ[R] n, rfl⟩
lemma le_comap_range_rTensor (q : Q) :
LinearMap.range g ≤ (LinearMap.range (rTensor Q g)).comap
((TensorProduct.mk R P Q).flip q) := by
| Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 101 | 105 |
/-
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.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | theorem liftPropWithinAt_self {f : H → H'} {s : Set H} {x : H} :
LiftPropWithinAt P f s x ↔ ContinuousWithinAt f s x ∧ P f s x :=
liftPropWithinAt_iff' ..
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 198 | 201 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | · exact left_mem_Ici
rename_i hi
obtain - | h' := chain'_iff_pairwise.mp h.1
exact (h' _ hi).le
· exact iSup₂_le fun i _ ↦ chainHeight_mono Set.inter_subset_left
| Mathlib/Order/Height.lean | 243 | 248 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
/-!
#... | hp.out.ne_zero, not_false_eq_true]
/-- This is the `n+1`st coefficient of our solution, projected from `root_exists`. -/
def succNthVal (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : Fin (n + 1) → k) (ha₁ : a₁.coeff 0 ≠ 0)
(ha₂ : a₂.coeff 0 ≠ 0) : k :=
| Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 104 | 108 |
/-
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.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... |
/-- constructor of a type defined by a qpf -/
def Fix.mk (x : F (Fix F)) : Fix F :=
Quot.mk _ (PFunctor.W.mk (q.P.map fixToW (repr x)))
/-- destructor of a type defined by a qpf -/
def Fix.dest : Fix F → F (Fix F) :=
Fix.rec (Functor.map Fix.mk)
| Mathlib/Data/QPF/Univariate/Basic.lean | 231 | 239 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.Data.Fin.FlagRange
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Dual.Basis
import Mathlib.RingTheory.SimpleR... | span_le.trans forall_mem_image
theorem flag_succ (b : Basis (Fin n) R M) (k : Fin n) :
b.flag k.succ = (R ∙ b k) ⊔ b.flag k.castSucc := by
| Mathlib/LinearAlgebra/Basis/Flag.lean | 42 | 45 |
/-
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.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | /-- The complement to a compact set belongs to a filter `f` if it belongs to each filter
`𝓝 x ⊓ f`, `x ∈ s`. -/
theorem IsCompact.compl_mem_sets (hs : IsCompact s) {f : Filter X} (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f) :
sᶜ ∈ f := by
contrapose! hf
| Mathlib/Topology/Compactness/Compact.lean | 48 | 52 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Sym.Basic
/-!
# Equivalence between `Multiset` and `ℕ`-valued finitel... | simp
| Mathlib/Data/Finsupp/Multiset.lean | 172 | 173 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | simp
@[reassoc (attr := simp)]
lemma XXIsoOfEq_inv_ιTotal {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ : x₂ = y₂)
(i₁₂ : I₁₂) (h : ComplexShape.π c₁ c₂ c₁₂ (x₁, x₂) = i₁₂) :
(K.XXIsoOfEq _ _ _ h₁ h₂).inv ≫ K.ιTotal c₁₂ x₁ x₂ i₁₂ h =
K.ιTotal c₁₂ y₁ y₂ i₁₂ (by rw [← h, h₁, h₂]) := by
| Mathlib/Algebra/Homology/TotalComplex.lean | 268 | 274 |
/-
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.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | edist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, edist (f i) (f (i + 1)) :=
Nat.Ico_zero_eq_range ▸ edist_le_Ico_sum_edist f (Nat.zero_le n)
/-- A version of `edist_le_Ico_sum_edist` with each intermediate distance replaced
| Mathlib/Topology/EMetricSpace/Basic.lean | 43 | 46 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.Lebesgu... | theorem hasCompactSupport_normed : HasCompactSupport (f.normed μ) := by
simp only [HasCompactSupport, f.tsupport_normed_eq (μ := μ), isCompact_closedBall]
| Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 75 | 77 |
/-
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... | Mathlib/Computability/NFA.lean | 158 | 164 | |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 367 | 371 | |
/-
Copyright (c) 2020 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.CardCommute
import Mathlib.Grou... | toFun
| .inl j => r j
| Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 125 | 126 |
/-
Copyright (c) 2020 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.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | theorem iteratedDeriv_eq_iterate : iteratedDeriv n f = deriv^[n] f := by
ext x
rw [← iteratedDerivWithin_univ]
convert iteratedDerivWithin_eq_iterate (F := F)
| Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 285 | 288 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Module.Projective
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
/-!
# ... | convert congr_arg Subtype.val (fg x)
replace si : LinearIndepOn K (fun x => f (g x)) s := by
simpa only [fg] using si.map' _ (ker_subtype _)
exact si.image_of_comp
· rintro ⟨s, hsc, si⟩
have : LinearIndepOn K f.rangeRestrict s :=
LinearIndependent.of_comp (LinearMap.range f).subtype (by ... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 107 | 126 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... | mem_support_iff.mpr ∘ trailingCoeff_nonzero_iff_nonzero.mpr
theorem natTrailingDegree_le_of_mem_supp (a : ℕ) : a ∈ p.support → natTrailingDegree p ≤ a :=
natTrailingDegree_le_of_ne_zero ∘ mem_support_iff.mp
| Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 250 | 253 |
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.ZMod.Basic
/-!
# Facts about factorials in ZMod
... | namespace ZMod
theorem cast_descFactorial {n p : ℕ} (h : n ≤ p) :
(descFactorial (p - 1) n : ZMod p) = (-1) ^ n * n ! := by
rw [descFactorial_eq_prod_range, ← prod_range_add_one_eq_factorial]
simp only [cast_prod]
nth_rw 2 [← card_range n]
rw [pow_card_mul_prod]
refine prod_congr rfl ?_
intro x hx
rw... | Mathlib/Data/ZMod/Factorial.lean | 31 | 42 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | · intro h
simp only [(hd :: tl).formPerm_apply_mem_eq_self_iff hl hd mem_cons_self,
add_le_iff_nonpos_left, length, nonpos_iff_eq_zero, length_eq_zero_iff] at h
simp [h]
| Mathlib/GroupTheory/Perm/List.lean | 330 | 333 |
/-
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.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... |
/-- Euler's product formula for the totient function. -/
theorem totient_eq_mul_prod_factors (n : ℕ) :
(φ n : ℚ) = n * ∏ p ∈ n.primeFactors, (1 - (p : ℚ)⁻¹) := by
by_cases hn : n = 0
· simp [hn]
have hn' : (n : ℚ) ≠ 0 := by simp [hn]
| Mathlib/Data/Nat/Totient.lean | 284 | 290 |
/-
Copyright (c) 2021 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.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | rw [lt_iff_le_and_ne, ← tsub_le_iff_right]
refine ⟨h.le, ?_⟩
| Mathlib/Algebra/Order/Sub/Defs.lean | 295 | 296 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | end LinearOrderedSemifield
| Mathlib/Algebra/Order/Field/Basic.lean | 697 | 698 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... |
theorem integral_one_div_of_pos (ha : 0 < a) (hb : 0 < b) :
∫ x : ℝ in a..b, 1 / x = log (b / a) := by simp only [one_div, integral_inv_of_pos ha hb]
theorem integral_one_div_of_neg (ha : a < 0) (hb : b < 0) :
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 456 | 460 |
/-
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.Module.Basic
import Mathlib.Algebra.Module.LinearMap.Defs
import Mathlib.RingTheory.HahnSeries.Basic
import Mathlib.Algebra.BigOperators.Group.... | y.coeff x.order = 0 := by
let xo := x.isWF_support.min (support_nonempty_iff.2 h)
have : xo = x.order := (order_of_ne h).symm
have hx : x.coeff xo = y.coeff xo + (single x.order x.leadingCoeff).coeff xo := by
nth_rw 1 [hxy, coeff_add]
| Mathlib/RingTheory/HahnSeries/Addition.lean | 220 | 224 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | instance : AddLeftMono NatOrdinal.{u} :=
⟨fun a _ _ h => nadd_le_nadd_left h a⟩
instance : AddLeftReflectLE NatOrdinal.{u} :=
⟨fun a b c h => by
by_contra! h'
exact h.not_lt (add_lt_add_left h' a)⟩
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 341 | 348 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | (h' : ∀ a (H : a ∈ x), f a H = none → x = none) : x.pbind f = none ↔ x = none := by
cases x
· simp
| Mathlib/Data/Option/Basic.lean | 166 | 168 |
/-
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 | 1,160 | 1,164 | |
/-
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... | lift_mk_eq.{v,u,w}.mpr ⟨(Equiv.ofInjective f hf).symm⟩
| Mathlib/SetTheory/Cardinal/Basic.lean | 664 | 665 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | theorem gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 :=
gauge_le_of_mem zero_le_one <| by rwa [one_smul]
/-- Gauge is subadditive. -/
theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
gauge s (x + y) ≤ gauge s x + gauge s y := by
refine le_of_forall_pos_lt_add fun ε hε =>... | Mathlib/Analysis/Convex/Gauge.lean | 175 | 181 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | instance : Sub ℍ[R,c₁,c₂,c₃] :=
| Mathlib/Algebra/Quaternion.lean | 264 | 264 |
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