chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
23,946 | lean_file_chunk | 714b3b044f409850ae3087089e06ecc35f3988f22672498a4cbd5f577ffd4cd7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li, Joachim Breitner
-/
module
public import Mathlib.Algebra.Order.Group.Int
public import Mathlib.Algebra.Order.SuccPred.WithBot
public import Mathlib.Data.ENat.... |
21,435 | lean_file_chunk | 5cae9f94a8741892ba2e1c27db153fae623dff12575be8e047c8a202a592f904 | lean_source_file | pkg_mathlib_source_file_chunks | rw [← krullDim_pos_iff, ← Nat.cast_zero, ← ENat.WithBot.add_one_le_iff, Nat.cast_zero, zero_add]
lemma krullDim_nonpos_of_subsingleton [Subsingleton α] : krullDim α ≤ 0 := by
rw [krullDim_nonpos_iff_forall_isMax]
exact fun x y h ↦ (Subsingleton.elim x y).ge
lemma krullDim_eq_zero [Nonempty α] [Subsingleton α] :
... |
14,454 | lean_file_chunk | 8e103705f18d7da455f2a7f1f3264e6d51856a917c620dc2dd5899a32555f3d5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Yaël Dillies
-/
module
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Tooling to make copies of lattice structures
Sometimes it is useful to ma... |
4,823 | lean_file_chunk | ded55fb65d1f9f46562471a489ebc2928d00b44cec1c445aa7582c856aaf01c8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Mathieu Guay-Paquet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mathieu Guay-Paquet
-/
module
public import Mathlib.Order.Ideal
/-!
# Order filters
## Main definitions
Throughout this file, `P` is at least a preorder, but some sections req... |
3,984 | lean_file_chunk | 4e464b127a9aafa6bef1b604a1c9ce042c641da0cb65887637681c0a9bd8f55d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
module
public import Mathlib.Order.BoundedOrder.Basic
public import Mathlib.Order.OmegaCompletePartialOrder
public import Mathlib.Order.ConditionallyComplete... |
12,962 | lean_file_chunk | d1359268ec300a0d2b29e36f50e886c8d6c13b05d8fa48121b47b3a5f2dc0dcf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
module
public import Mathlib.Data.Nat.Lattice
public import Mathlib.Logic.Denumerable
public import Mathlib.Logic.Function.Iterate
public import Mathlib.Order.Hom.Ba... |
14,269 | lean_file_chunk | 944ec85232a9c4c6408b48019a616ad3ae41ccfebddacfe24a88051450777bec | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Tactic.Linter.DeprecatedModule
public import Mathlib.Order.RelClasses
public import Mathlib.Order.Interval.Set.B... |
1,796 | lean_file_chunk | 845872d59843ab228fa9d2935e082b8693b2f87177e0be36cf9776d3c57ecafa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Simon Hudon, Yury Kudryashov
-/
module
public import Mathlib.Order.Notation
/-!
# Order-related type synonyms
In this file we define `WithBot` and `WithTop`.
-/
@[e... |
13,939 | lean_file_chunk | 15c5d1360cf321e8128f5293e974a7875c99fb6bd1f1fd24923e307d641585c1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
module
public import Mathlib.Data.Set.Image
public import Mathlib.Util.Delaborators
/-!
# Directed indexed families and sets
This file defines direct... |
17,414 | lean_file_chunk | fd936390ab3c7699618008334ce321bc4f3f911676d44d78b99f6f12ad7efe96 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Yaël Dillies
-/
module
public import Mathlib.Data.Set.Monotone
public import Mathlib.Order.Cover
public import Mathlib.Order.LatticeIntervals
public import Mathlib.Ord... |
23,986 | lean_file_chunk | 32f34bd4b4f4828915a4a4063e98a210dbad4574c34534e65f4c86dd0d681aee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
module
public import Mathlib.Control.Monad.Basic
public import Mathlib.Dynamics.FixedPoints.Basic
public import Mathlib.Order.CompleteLattice.Basic
public i... |
6,948 | lean_file_chunk | 9d01aaeb2afd2d6fa2c80949a7840bc5f818eaa82be327fbe77deb92977b4437 | lean_source_file | pkg_mathlib_source_file_chunks | def id : α →𝒄 α := ⟨OrderHom.id, ωScottContinuous.id.map_ωSup⟩
/-- The composition of continuous functions. -/
@[simps!]
def comp (f : β →𝒄 γ) (g : α →𝒄 β) : α →𝒄 γ :=
⟨.comp f.1 g.1, (f.ωScottContinuous.comp g.ωScottContinuous).map_ωSup⟩
@[ext]
protected theorem ext (f g : α →𝒄 β) (h : ∀ x, f x = g x) : f = g... |
15,756 | lean_file_chunk | c1d5dd4eb66190141dfcec14bbf7dfdcda01ef25254fc2d21101d7d28fa0cc1e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Order.Lattice
public import Mathlib.Tactic.Order
/-!
# Circular order hierarchy
This file defines circular preorders, circular partial ord... |
7,905 | lean_file_chunk | 05dfcfa14aac7834e9705c034331c4a150dc76851fc5c68b19a5c12b2fa7d2e4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Data.Set.Operations
public import Mathlib.Util.Notation3
/-!
# Notation classes for set supremum and i... |
8,898 | lean_file_chunk | e20e3b12bca9f3802544a86bf29113434ebc07a1395ea305f82e7afe5eb06d38 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and... |
5,822 | lean_file_chunk | 285fc6b56d36e6d259a0fc3c39014a0b4fd8965fec6981e2f804b0b469101621 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.Group.Units.Equiv
public import Mathlib.Algebra.Order.Group.End
public import Mathlib.Logic.Function.Conjugate
public import M... |
10,807 | lean_file_chunk | 6a199fd584ae0fbf7680398a022fbe7ab66c6b3bd901bda7d0b0359f26b7de6a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Krause. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chriara Cimino, Christian Krause
-/
module
public import Mathlib.Order.Closure
public import Mathlib.Order.Hom.CompleteLattice
/-!
# Nucleus
Locales are the dual concept to frames... |
16,916 | lean_file_chunk | ef4af72f10e12d3f56209cd3ef82172ca3ed74fee98a37c5c800f1aff9ab0d24 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Logic.Relation
public import Mathlib.Order.Hom.Basic
public import Mathlib.Tactic.Tauto
/-!
# Turning a preorder into a partial order
This... |
7,281 | lean_file_chunk | a2130e1b1219068012272d98b2d3887faa62743baca15b98a90873233fe254ae | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin, Violeta Hernández Palacios
-/
module
public import Mathlib.Order.CompleteLattice.Defs
public import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Set.Lat... |
19,826 | lean_file_chunk | 410725a611c57e416f2bf546404f66df43dd7f2f059efb83d4d3b909abd6661f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Order.Sublattice
/-!
# Boolean subalgebras
This file defines Boolean subalgebras.
-/
@[expose] public section
open Function Set
variabl... |
23,959 | lean_file_chunk | b908a908a58b52d138db7db905263f6f77d62b82d52394bdfa11296a6dd02d81 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Data.Bool.Basic
public import Mathlib.Logic.Pairwise
public import Mathlib.Order.Monotone.Basic
public import Mathlib.Order.ULift
/-!
#... |
14,520 | lean_file_chunk | 5954f78bcefe989a452948697db2c95da9e3f1455dd253cee89b8c688afa9ccd | lean_source_file | pkg_mathlib_source_file_chunks | protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
@[to_dual map_inf_le]
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (... |
9,649 | lean_file_chunk | 905fd8227f91ac22327b4c0bcd12592860888b03bd38655cc8c50ef6d5137e96 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Yury Kudryashov
-/
module
public import Mathlib.Dynamics.FixedPoints.Basic
public import Mathlib.Order.Hom.Order
public import Mathlib.Order.OmegaCompletePa... |
6,898 | lean_file_chunk | 7dcf053a7f81a545c9e55dce505c51b8cbc4513736554618e3fe2dd38522942f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Vlad Tsyrklevich
-/
module
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Data.Set.Card
/-!
# Maximal length of chains
This file contains lemmas to work wi... |
23,968 | lean_file_chunk | bbbce8d753a00eb4331630fef285426d0237daff35eb20201134e721ba5ef405 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
module
public import Mathlib.Data.Finset.Lattice.Prod
public import Mathlib.Data.Finset.Powerset
public import Mathlib.Data.Set.F... |
2,613 | lean_file_chunk | a4ab9e0f3c2a2c55aa267e521c5bc135eb115d736ed30311b304eaaf715819d1 | lean_source_file | pkg_mathlib_source_file_chunks | (hf : ∀ i, f i ∈ s) : ⨅ i, f i ∈ s := hs.dual.iSup_mem_of_nonempty hf
lemma SupClosed.sSup_mem_of_nonempty (hs : SupClosed s) (ht : t.Finite) (ht' : t.Nonempty)
(hts : t ⊆ s) : sSup t ∈ s := by
have := ht.to_subtype
have := ht'.to_subtype
rw [sSup_eq_iSup']
exact hs.iSup_mem_of_nonempty (by simpa)
lemma I... |
9,613 | lean_file_chunk | 1ac71b24bd27c4c25f31f611c0f75ed7e6c3c9bcc7fa1daa63251e5d42769cb7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov, Yaël Dillies
-/
module
public import Mathlib.Order.OrderDual
/-!
# Minimal/maximal and bottom/top elements
This file defines predicates for elements t... |
9,106 | lean_file_chunk | acfb165e04986f1aeb28cb7a5c852bcd92af8d5fb4d9e22eebd03827134f2074 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Order.Bounds.Basic
/-!
# Intervals in Lattices
In this file, we provide instances of lattice structures on intervals within lattices.
... |
8,933 | lean_file_chunk | 97a761bc0639b851c3148493bbe724e7e8a2d1f43fe6bacbcd3b8d84ad63de6b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Finn Mortimore. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Finn Mortimore
-/
module
public import Mathlib.Order.Preorder.Chain
public import Mathlib.Data.Set.Lattice
public import Mathlib.Dynamics.FixedPoints.Basic
public import Mathlib.Order... |
1,626 | lean_file_chunk | 4abc3d6560ac9f8a9a724350b41e6180f3a0b5decbc7a1ea186b1b43f4366b65 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Colva Roney-Dougal. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Colva Roney-Dougal, Inna Capdeboscq, Susanna Fishel, Kim Morrison
-/
module
public import Mathlib.Order.Atoms
/-!
# The radical of a lattice
This file contains results on the or... |
13,604 | lean_file_chunk | 9e90e37073ac2e3bbb8bba1003692b3be6a1d74b56ca9b6eb722c8920a007297 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Order.Lex
public import Mathlib.Order.WellFounded
public import Mathlib.Tactic.Common
/-!
# Lexicographic order on Pi types
This file defi... |
21,292 | lean_file_chunk | 9e41333371a9f4259bdcdda8b1c6148ebc72dae7ec21bbf1fe1a86274501026a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Gabin Kolly. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly, David Wärn
-/
module
public import Mathlib.ModelTheory.DirectLimit
public import Mathlib.Order.Ideal
/-!
# Partial Isomorphisms
This file defines partial is... |
5,965 | lean_file_chunk | 7446e808e28a84bac0a8321922f400bce58e8cf9c665c4367a1dc041e8a13912 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.ElementarySubstructures
/-!
# Skolem Functions and Downward Löwenheim–Skolem
## Main Definitions
- `FirstOrder.Language.s... |
19,826 | lean_file_chunk | 9b8da7bf6569d03d62f9b5c443c386f19876dda0fa7443d0714f78d51dc2819c | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.ModelTheory.FinitelyGenerated
public import Mathlib.ModelTheory.PartialEquiv
public import Mathlib.ModelTheory.Bundled
publ... |
23,976 | lean_file_chunk | cee873b32899c0cb286528a0d343f6825caed7e1fcc764a6e1a30d767926cd15 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Data.Finset.Basic
public import Mathlib.ModelTheory.Syntax
pub... |
16,623 | lean_file_chunk | afb1ccff5a9d8e30cca368a8859645dbe47129d6ca5c16c2a74a1c3f369b6026 | lean_source_file | pkg_mathlib_source_file_chunks | nonrec def Sentence.Realize (φ : L.Sentence) : Prop :=
φ.Realize (default : _ → M)
-- input using \|= or \vDash, but not using \models
@[inherit_doc Sentence.Realize]
infixl:51 " ⊨ " => Sentence.Realize
@[simp]
theorem Sentence.realize_not {φ : L.Sentence} : M ⊨ φ.not ↔ ¬M ⊨ φ :=
Iff.rfl
namespace Formula
@[sim... |
3,641 | lean_file_chunk | f63283f14b93f2a63646b96009c8d5fc2846c763a0cebeea3526bfd05982bdd3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.Satisfiability
public import Mathlib.Combinatorics.SimpleGraph.Basic
/-!
# First-Order Structures in Graph Theory
This fil... |
19,448 | lean_file_chunk | ec1ddf1eae69b778acd317613c80609fc47fe236ddb32d66b85c1d0bc5633490 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.Equivalence
/-!
# Quantifier Complexity
This file defines quantifier complexity of first-order formulas, and constructs pr... |
8,263 | lean_file_chunk | b4fc35999e6fb2d2de10d430a94d23e2138314cdbdc47ddc3bd4fbb8bdba4a0d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.ElementaryMaps
public import Mathlib.ModelTheory.Definability
/-!
# Elementary Substructures
## Main Definitions
- A `Fir... |
2,622 | lean_file_chunk | b0f396754802759c4b97fce9ad5f4bea84db8b07d876261d419bd855aeba0e53 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Data.Fintype.Quotient
public import Mathlib.ModelTheory.Semantics
/-!
# Quotients of First-Order Structures
This file defines prestruc... |
18,754 | lean_file_chunk | 6288a7969680e514e203e3f8ea26642b829cc23a5ade0579a83f0a1d3c195d6f | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.ModelTheory.Basic
/-!
# Language Maps
Maps between first-ord... |
11,695 | lean_file_chunk | ff49903a86ab0d55f178760e2f6003fa9ba6523fdaebd241e530dcfc88085291 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Data.Set.Finite.Lemmas
public import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defin... |
22,148 | lean_file_chunk | dde74b59aad17a1d50fde8e1e6a934bd765ecedc4861fc35693a8704c28d5677 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Algebra.CharZero.Infinite
public import Mathlib.Data.Rat.Encodable
public import Mathlib.Data.Finset.Sort
public import Mathlib.ModelThe... |
11,014 | lean_file_chunk | 870d544aa93d8041fc4d3280598a3702a2351ac68e69c3242bfbc7c048343c82 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Data.Fintype.Basic
public import Mathlib.ModelTheory.Substructures
/-!
# Elementary Maps Between First-Order Structures
## Main Defini... |
20,615 | lean_file_chunk | af4e9ef3314b4b9fcf71757477028f59ce48172153dba1b0621fd9e4a90bc752 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Data.Finite.Sum
public import Mathlib.Data.Fintype.Order
public import Mathlib.ModelTheory.FinitelyGenerated
public import ... |
23,981 | lean_file_chunk | b834e29010f6d1c66fac42d067a122cc2a0929e977d1a9501165a87c365c3262 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Data.Set.Prod
public import Mathlib.Logic.Equiv.Fin.Basic
publ... |
10,183 | lean_file_chunk | 6d9662786b9aacd38ce1d1a7c3584ff0ae768379071dc92bc4a0813c605af5e5 | lean_source_file | pkg_mathlib_source_file_chunks | let _ := Fintype.ofFinite β
((Finset.univ : Finset β).toList.map f).foldr (· ⊓ ·) ⊤
end BoundedFormula
namespace LHom
open BoundedFormula
/-- Maps a bounded formula's symbols along a language map. -/
@[simp]
def onBoundedFormula (g : L →ᴸ L') : ∀ {k : ℕ}, L.BoundedFormula α k → L'.BoundedFormula α k
| _k, falsu... |
23,958 | lean_file_chunk | d048f21c7c869933fed5fa0c147d2d6cc8919cffd0eb2d7fbf7965879328e453 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on First-Order Structur... |
4,844 | lean_file_chunk | fbf6939cc43ef52c0bc08c7ea80b76894506d0797f73ba5b236fd089f3343646 | lean_source_file | pkg_mathlib_source_file_chunks | rw [← comp_toEmbedding, symm_comp_self, refl_toEmbedding]
@[simp]
theorem self_comp_symm_toEmbedding (f : M ≃[L] N) :
f.toEmbedding.comp f.symm.toEmbedding = Embedding.refl L N := by
rw [← comp_toEmbedding, self_comp_symm, refl_toEmbedding]
@[simp]
theorem symm_comp_self_toHom (f : M ≃[L] N) :
f.symm.toHom.... |
23,921 | lean_file_chunk | c30e1b3940d31ae32234ef943a76a1abc2e5d93c5d4007e4ef2733fa16f572b3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.Ultraproducts
public import Mathlib.ModelTheory.Bundled
public import Mathlib.ModelTheory.Skolem
public import Mathlib.Order... |
6,567 | lean_file_chunk | 703a4b2f4eca5e24f5520e5f3b772fd97c8a3efc453f6412ea3591a29e30890b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.ElementarySubstructures
public import Mathlib.CategoryTheory.ConcreteCategory.Bundled
/-!
# Bundled First-Order Structures
... |
23,981 | lean_file_chunk | 052e4b86af0dbd933f0d48af8b4c333b48661e507667ff716fc1fd59f025730f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
module
public import Mathlib.Data.Fintype.Order
public import Mathlib.Order.Closure
public import Mathlib.ModelTheory.Semantics
public import Mathlib.Mo... |
9,718 | lean_file_chunk | 6e41662ea745a70d3bec371387eb08b280e75f3a58376130c88165055f8f1338 | lean_source_file | pkg_mathlib_source_file_chunks | /-- Reduces the language of a substructure along a language hom. -/
def substructureReduct (φ : L →ᴸ L') [φ.IsExpansionOn M] :
L'.Substructure M ↪o L.Substructure M where
toFun S :=
{ carrier := S
fun_mem := fun {n} f x hx => by
have h := S.fun_mem (φ.onFunction f) x hx
simp only [LHom.m... |
13,491 | lean_file_chunk | 1a8f84dd53eba6a7877a28d298252f74f56e0b4955513cf92ac42d3b108b4a1b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Computability.Encoding
public import Mathlib.Logic.Small.List
public import Mathlib.ModelTheory.Syntax
public import Mathlib.SetTheory.C... |
6,109 | lean_file_chunk | 986e6ce13bf3349b2ccce0b24517b3a63c1c61461820489c70ea3441a749bac0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.Quotients
public import Mathlib.Order.Filter.Finite
public import Mathlib.Order.Filter.Germ.Basic
public import Mathlib.Orde... |
8,952 | lean_file_chunk | 2461967cbcee136ca0e890c5810bddf02b934880cff39cdccc06ee060a608081 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.Satisfiability
/-!
# Type Spaces
This file defines the space of complete types over a first-order theory.
(Note that types... |
8,809 | lean_file_chunk | 3918a435903ddbb94bb75116b0e3ad72ef7b96254aab296b3f1e6c7f63a24267 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.ModelTheory.Satisfiability
/-!
# Equivalence of Formulas
## Main Definitions
- `FirstOrder.Language.Theory.Imp`: `φ ⟹[T] ψ` indicates ... |
23,907 | lean_file_chunk | d3d2476e379f3cd677d2dd14870196712f8a2c313e4373b83df5763ec0ea3ae3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Data.SetLike.Basic
public import Mathlib.Data.Rel
public import Mathlib.ModelTheory.Semantics
public import Mathlib.Tactic.FunProp
/-!
... |
1,457 | lean_file_chunk | 7ea50bc1ce9917e5479e96730b203a93d530eb32731aaa05bb52add3c77af272 | lean_source_file | pkg_mathlib_source_file_chunks | /-- Constant functions are `TermDefinable`, assuming the constant value is a language constant. -/
@[fun_prop]
theorem TermDefinable.const (C : L[[A]].Constants) : A.TermDefinable L (Function.const (α → M) C) :=
⟨C.term, by simp only [Term.realize_constants]; rfl⟩
/-- Constant functions are `TermDefinable₁`, assumin... |
7,902 | lean_file_chunk | 1759543b451f1c200f823e16cac089e38bef4c0614859a2f261bf1c85b14bdd5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Defs
public import Mathlib.Algebra.Module.LinearMap.Defs
public import Mathlib.Algebra.Order.Hom.Ring
... |
115 | lean_file_chunk | d4c37d7a92f33169b7b8183b17ad01021e0211df5a4608287a8ad2f1dfd6bf6d | lean_source_file | pkg_mathlib_source_file_chunks | module -- shake: keep-all
public import Mathlib.Topology.Covering.Basic
deprecated_module (since := "2025-12-10") |
2,191 | lean_file_chunk | ea6d1b610c7ec57d27a6d577a25889af846e12cbe49a0012edbbec2652d5baf2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.UrysohnsLemma
public import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Urysohn's lemma for bounded continuous functi... |
20,456 | lean_file_chunk | 5f0042f402e81da4a682140315a5e8ea83daab060b08ec52e07e10c3a4ce7418 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
module
public import Mathlib.Algebra.Order.Interval.Set.Instances
public import Mathlib.Order.Interval.Set.ProjIcc
public import Mathlib.Topology.Algeb... |
23,963 | lean_file_chunk | 108664f15e1197829bcc2d071034be7a529b22a21799660db0b5c380fd35e215 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
module
public import Mathlib.Topology.Hom.ContinuousEval
public import Mathlib.Topology.ContinuousMap.Basic
public import Mathlib.Topology.Separation.Regular
/-!
# The co... |
1,969 | lean_file_chunk | 397448d7d5c7a060e9c49a5759158b7d5d909624efe6ea7da319637812154e9d | lean_source_file | pkg_mathlib_source_file_chunks | def curry [LocallyCompactSpace X] [LocallyCompactSpace Y] : C(X × Y, Z) ≃ₜ C(X, C(Y, Z)) :=
⟨⟨ContinuousMap.curry, uncurry, by intro; ext; rfl, by intro; ext; rfl⟩,
continuous_curry, continuous_uncurry⟩
/-- If `X` has a single element, then `Y` is homeomorphic to `C(X, Y)`. -/
def continuousMapOfUnique [Unique X... |
23,961 | lean_file_chunk | 097831334eb47df4b44ad6e7ba1dd4a69ba47366212912a7f0c7515795a163f0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.Order.Group.Indicator
public import Mathlib.Analysis.Normed.Affine.AddTorsor
public import Mathlib.Analysis.Normed.Group.Funct... |
1,208 | lean_file_chunk | abc252e48dc43f0932af95345c69bb43d4da001eba8741ce4e145d7580a82f33 | lean_source_file | pkg_mathlib_source_file_chunks | `0 ≤ f ≤ 1`, `f = 1` on K and the support of `f` is contained in `V`. -/
lemma exists_continuousMap_one_of_isCompact_subset_isOpen [R1Space X] [LocallyCompactSpace X]
{K V : Set X} (hK : IsCompact K) (hV : IsOpen V) (hKV : K ⊆ V) :
∃ f : C(X, ℝ), Set.EqOn f 1 K ∧ IsCompact (tsupport f) ∧
tsupport f ⊆ V ∧ ... |
17,667 | lean_file_chunk | fde7fc80cf45fa89a1764f2096998e010065fcfbef7d677c9f77152e1596213f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
module
public import Mathlib.Topology.Bases
public import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines... |
7,443 | lean_file_chunk | a0e37b12eab858d38b9e2835e75e369f7854fb2289f62f7f1b989c2b2e83c621 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Topology.Constructions
public import Mathlib.Tactic.TFAE
/-!
# Locally closed sets
## Main definitions
* `IsLocallyClosed`: Predicate sayin... |
4,422 | lean_file_chunk | 1e1711f080cd67303b0a5acffd20145d3fc3d5bc8d93f0f4be533b1707404bd3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Order.Category.Preord
public import Mathlib.Topology.Category.TopCat.Basic
public import Mathlib.Topology.ContinuousMap.Basic
public import ... |
23,977 | lean_file_chunk | d8cb8fc9bfe5e38dd60d3c413e8ee704b82f3c374c3b113eee0c13cfb4a164a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Data.Set.Constructions
public import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
public import Mathlib.Topology.Con... |
23,936 | lean_file_chunk | bce031ad6fc0946007ff48ea0aea8f2e753f847c94b708db27fada395ca76075 | lean_source_file | pkg_mathlib_source_file_chunks | latter, use `TopologicalSpace.SeparableSpace s` or
`TopologicalSpace.IsSeparable (univ : Set s)`). In metric spaces, the two definitions are
equivalent, see `TopologicalSpace.IsSeparable.separableSpace`. -/
def IsSeparable (s : Set α) :=
∃ c : Set α, c.Countable ∧ s ⊆ closure c
theorem IsSeparable.mono {s u : Set α}... |
7,608 | lean_file_chunk | 42f8fd0c02bce832e563f4c53f3c9d3c76526b20df19cfcccce89a5e3fb41aaf | lean_source_file | pkg_mathlib_source_file_chunks | obtain ⟨u, u_mem, xu, uv⟩ : ∃ u ∈ countableBasis α, x ∈ u ∧ u ⊆ v :=
(isBasis_countableBasis α).isOpen_iff.1 hv _ hx
have : x ∈ ⋃ a ∈ s u, a := by
convert xu
exact (hs u u_mem).symm
obtain ⟨w, ws, xw⟩ : ∃ w ∈ s u, x ∈ w := by simpa using this
refine ⟨w, ⟨u, u_mem, ws⟩, xw, ?_⟩
apply Su... |
2,546 | lean_file_chunk | 28e9e0fac1597282cabcdbc590fff58a4d8cac6425b2d874d2f579def4616a3c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.Topology.LocallyConstant.Basic
public import Mathlib.Logic.Function.FiberPartition
/-!
This file provides some API surrounding `Fun... |
6,330 | lean_file_chunk | 1069d0b63453cd4f87941b2bb77ea03cc033f61afe7ec86a2cb438c7ed3dcb73 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Data.Set.BoolIndicator
public import Mathlib.Topology.ContinuousOn
/-!
# Clopen sets
A clopen set is ... |
15,610 | lean_file_chunk | a5f13c20d6097e680f3d5c5807b4ade7739754833737170619d6a5ea83168f31 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Topology.Neighborhoods
/-!
# Lemmas on cluster and accumulation points
In this file we prove various le... |
4,287 | lean_file_chunk | 1c7f5cf2cece5a89cdf604c086896f1887928322c7a2655fd4c4355ec8638d3b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Daniel Weber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Weber
-/
module
public import Mathlib.Topology.Perfect
public import Mathlib.Tactic.Peel
/-!
# Derived set
This file defines the derived set of a set, the set of all `AccPt`s o... |
8,760 | lean_file_chunk | 2e323ab6ae5e023a69f66966b8b055c39b2765e76bc10bbc6d15d13c564967b4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Topology.Constructions
public import Mathlib.Order.Filter.ListTraverse
public import Mathlib.Tactic.AdaptationNote
public import Mathlib.To... |
8,956 | lean_file_chunk | 8aef21f1da7276fdeeb6986396c52fb7bc2f570f0e508879e75ba4298830dbae | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Topology.LocalAtTarget
public import Mathlib.Topology.Separation.Regular
public import Mathlib.Tactic.StacksAttribute
/-!
# Jacobson spaces
... |
7,681 | lean_file_chunk | 7263f829b2a0c086d49c1f750ec234b1cf32aa8e350f06e11f07dbfdad48a45c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.ContinuousOn
/-!
### Continuity of piecewise defined functions
-/
public section
open Set Filter Function Topology Filter
v... |
23,943 | lean_file_chunk | 9630d6c40e48e2f878f11e1fbadfbe82cc1a4016e025e1a203b01a7975b44a79 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
module
public import Mathlib.Order.UpperLower.Closure
public import Mathlib.Order.UpperLower.Fibration
public import Mathlib.Tactic.TFAE
public import Mat... |
8,153 | lean_file_chunk | e8910a881b37983d9cd0279e89e6ee36a4809c1cec4c3f56b2db697ecadf395c | lean_source_file | pkg_mathlib_source_file_chunks | Quotient.subsingleton_iff.trans inseparableSetoid_eq_top_iff
instance [TopologicalSpace α] [IndiscreteTopology α] : Subsingleton (SeparationQuotient α) :=
subsingleton_iff.2 ‹_›
theorem nontrivial_iff [TopologicalSpace α] :
Nontrivial (SeparationQuotient α) ↔ NontrivialTopology α := by
simpa [not_subsingleton... |
2,458 | lean_file_chunk | 558d4e6d800787d968b4a888b32238abfe9df6b7d4bb90958f86d2433a5208c4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
module
public import Mathlib.Order.Filter.Partial
public import Mathlib.Topology.Neighborhoods
/-!
# Partial functions and topological spaces
In this file we prove p... |
11,116 | lean_file_chunk | c7b003b671a3a0deb7ffe3eedf0a225e03d9a46d5950a12486413ec09a5db974 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Topology.Connected.Basic
public import Mathlib.Topology.Separation.Hausdorff
public import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps ... |
10,077 | lean_file_chunk | 26f2cb3d2546d5898b301462bf74c504f8608faf1763945f83fe04011fd8339d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
module
public import Mathlib.Topology.Separation.Regular
/-!
# Perfect Sets
In this file we define perfect subsets of a topological space, and prove some basic p... |
5,617 | lean_file_chunk | e313d6fd2c5e567a76fcfd9d2f6db9662b47a2877c31dd7c1896f384cdfbd936 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Topology.Separation.Basic
/-!
# Pointwise convergence of indicator functions
In this file... |
23,585 | lean_file_chunk | 061d28d4c3665ce6bfecf5278473213009737ffff3b6b95a30af92031a62f774 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Order.Filter.Lift
public import Mathlib.Topology.Basic
/-!
# Interior, closure and frontier of a set
Th... |
23,986 | lean_file_chunk | d5fffb493cc519335409de0f6033a4478b5d2bc87b3d76e1222154856dcf3592 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Topology.Continuous
public import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies... |
18,602 | lean_file_chunk | 0a47b7410963204bc5b38ebb56edb2848a547ffb4e884b672a46566f4927b7f0 | lean_source_file | pkg_mathlib_source_file_chunks | continuous_of_discreteTopology
theorem le_generateFrom {t : TopologicalSpace α} {g : Set (Set α)} (h : ∀ s ∈ g, IsOpen s) :
t ≤ generateFrom g :=
le_generateFrom_iff_subset_isOpen.2 h
theorem induced_generateFrom_eq {α β} {b : Set (Set β)} {f : α → β} :
(generateFrom b).induced f = generateFrom (preimage f ... |
12,953 | lean_file_chunk | 8f47ee247084880ec8955170c3deeb2f8dd2ad59010246d554e4f1fba297e377 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.Topology.OpenPartialHomeomorph.Composition
public import Mathlib.Topology.SeparatedMap
/-!
# Local homeomorphisms
This file defines ... |
3,802 | lean_file_chunk | a22cc3f9942f7e7da4b34bc41ff6d8bc7c8742923fc6c4857ba5aa9397df4109 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.Defs.Sequences
public import Mathlib.Topology.ContinuousOn
/-!
# Topology generated by its restrictions to subsets
We say t... |
16,150 | lean_file_chunk | 3a054177aa2a61ab3113c4496486b41786395ae34d25beb9d9eba2bed152f37e | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Order.Filter.AtTopBot.Basic
public import Mathlib.Topology.Closure
/-!
# Neighborhoods in topological sp... |
8,974 | lean_file_chunk | 9e8188dfb66d031838bcd2f38be28ea9696ddbc42f0dc002b25aa387f6e53c4a | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Data.Set.Finite.Basic
public import Mathlib.Data.Set.Finite.Range
public import Mathlib.Data.Set.Lattice
... |
10,046 | lean_file_chunk | f1bfea4dd059bee6668891b44f41ea1ad04653a43527749fb2788599810a5109 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Data.Set.Image
public import Mathlib.Topology.Bases
public import Mathlib.Topology.Inseparable
public import Mathlib.Topology.Compactness.Nh... |
18,505 | lean_file_chunk | 0399f554575a652c0e95bc6bda9dc104e7d9114c1a73f217f576dc54af426b8f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.CategoryTheory.GlueData
public import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
public import Mathlib.Topology.Category.TopCat.Opens
p... |
242 | lean_file_chunk | 1ff8603868bb1625aca01d99a59001c83382c1d147e91f3cce1113a1008919ef | lean_source_file | pkg_mathlib_source_file_chunks | module -- shake: keep-all
import Mathlib.Algebra.Order.Module.Field
import Mathlib.Data.EReal.Operations
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.MetricSpace.Bounded
deprecated_module (since := "2025-12-17") |
8,553 | lean_file_chunk | 82c44567ec32e997a3c05d19f63f7bfe8eefc970ae60ad721665010e01f8e507 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Order.Filter.Lift
public import Mathlib.Order.Interval.Set.Monotone
public import Mathlib.Topology.Separation.Basic
/-!
# Topology on... |
23,971 | lean_file_chunk | 7086bd175b6f277fe74a4edf8d711bc09b0d4a7e5e5827151d8c6dc188c9c560 | lean_source_file | pkg_mathlib_source_file_chunks | /-
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
-/
module
public import Mathlib.Algebra.Group.TypeTags.Basic
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Data.Finse... |
23,956 | lean_file_chunk | fedca7bf66f3153d49f2fc6e8dc25a436fb2d247ebbabf6b6849aafe21f350bb | lean_source_file | pkg_mathlib_source_file_chunks | a continuous injective map is also discrete. -/
theorem DiscreteTopology.preimage_of_continuous_injective {X Y : Type*} [TopologicalSpace X]
[TopologicalSpace Y] (s : Set Y) [DiscreteTopology s] {f : X → Y} (hc : Continuous f)
(hinj : Function.Injective f) : DiscreteTopology (f ⁻¹' s) :=
DiscreteTopology.of_c... |
10,236 | lean_file_chunk | 0e43e454ebf4ed1307ee258a04c7d401d09617c7466ed3d736d30857fac95d68 | lean_source_file | pkg_mathlib_source_file_chunks | where `Π i, A i` is endowed with the usual product topology. -/
theorem inducing_iInf_to_pi {X : Type*} (f : ∀ i, X → A i) :
@IsInducing X (∀ i, A i) (⨅ i, induced (f i) inferInstance) _ fun x i => f i x :=
letI := ⨅ i, induced (f i) inferInstance; ⟨(induced_to_pi _).symm⟩
variable [Finite ι] [∀ i, DiscreteTopol... |
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