mathlib : Leans mathematical library Johannes Hlzl VU Amsterdam - PowerPoint PPT Presentation

mathlib : Lean’s mathematical library Johannes Hölzl VU Amsterdam Lean Together 2019 Amsterdam

Introduction — what is mathlib (classical) mathematical library for Lean classical – linear algebra, number theory, analysis, ... classical – using choice and LEM Formerly distributed with Lean itself Leo wanted more flexibility Some (current) topics: (Linear) Algebra, Analysis, Set Theory, Number Theory, ... Goal: be comparable to Coq’s mathematical components, Isabelle’s HOL-Analysis 1 23

Development 120 , 000 100 , 000 Lines of lean Code 80 , 000 60 , 000 40 , 000 20 , 000 0 n b r r y n l g p t v c n a p u c a o e a u u e a e J O M A M D J F A S N J J Month in 2018 Repository starts July 2017 with 22.000 LoC 2 23

Organization Constructions Tactics Theories 3 23

Organization

B asic parts tactic mathlib ’s tactics logic Lemmas about logical connectices; classical function inverse; Schröder-Bernstein; ... order Lemmas about orders; further order and lattice type classes algebra Lemmas about algebraic structures; more algebraic and order type classes, upto modules category Type category (Haskell style): Functor, Applicative, Monads, Traversable data Generic (mostly computable) types 4 23

T heories set_theory Ordinals, Cardinals, a model of ZFC category_theory Category theory: functors, natural trans., limits, ... linear_algebra Modules, Vector spaces, Tensor products, Dimensions, ... analysis Analysis, Topology, Measure theory, ... group_theory Free (abelian) groups, Order of an element, ... ring_theory Principal ideal domains, ... field_theory Finite fields, perfect closure, ... computability Turing machines etc. number_theory Diophantine equations, Pell’s equation (?) 5 23

data – Generic (mostly computable) types � n nat N , gcd , mod , prime , , ... � p int Z (as datatype, not quotient) rat Q (as datatype, not quotient) real R (as quotient over Cauchy sequences) equiv Isomorphism between Type fin fin n = { 0 ,..., n − 1 } set set α := α → Prop list list α multiset multiset α := list α / perm finset finset α := { m : multiset α | nodup m } ... 6 23

C onstructions

C onstructions – Ordered groups and rings nonneg _ comm _ group , nonneg _ ring and linear _ nonneg _ ring Helper type classes to construct ordered _ group and ordered _ ring Specify nonneg : α → Prop and (optionally) pos : α → Prop Constructs the order on the group / ring Proves the relation btw order and algebraic operations 7 23

C onstructions – Adjoin ⊤ , ⊥ or 0 with _ top , with _ bot and with _ zero := option Ext. nonnneg. reals: R ≥ 0 ⊎ ∞ , extended nats N = N ⊎ ∞ Multisets with infinity (e.g. sets of factors, ∞ to represent 0) Example instances: ◮ partial _ order α → partial _ order ( with _ top α ) ◮ add _ monoid α → add _ monoid ( with _ top α ) ◮ canonically _ ordered _ comm _ semiring α → canonically _ ordered _ comm _ semiring ( with _ top α ) 8 23

S tructures as Functors and Complete Lattices There are many set like structures: filter , topology , uniformity , submodule , measurable Form complete lattices, e.g. topological _ space α (Co)functors, e.g. map : ( α → β ) → ( filter α → filter β ) Exception: uniformity only cofunctor Order + comap is usually the morphism definition Used as light-weight category theory (simpler to setup, base constructions require less setup) 9 23

Example: Topologies structure topo ( α : Type) := (is_open : set α → Prop) ... -- univ, ∩ , � cont : ( α → β ) → topo α → topo β → Prop cont f T 1 T 2 : ↔ ∀ s, is_open T 2 s → is_open T 1 (f − 1 ’s) ( ≤ ) : topo α → topo α → Prop T 1 ≤ T 2 : ↔ ∀ s, is_open T 1 s → is_open T 2 s map : ( α → β ) → (topo α → topo β ) − 1 ’ s) is_open (map T f) s : ↔ is_open T (f cont f T 1 T 2 : ↔ T 2 ≤ T 1 .map f 10 23

Example: Generating Topologies generate : set (set α ) → topo α ∀ s ∈ g, is_open (generate g) s ∀ T, ( ∀ s ∈ g, is_open T s) → generate g ≤ T map : ( α → β ) → (topo α → topo β ) − 1 ’ s) is_open (map T f) s : ↔ is_open T (f comap : ( α → β ) → (topo β → topo α ) is_open (comap T f) s : ↔ ∃ t, is_open T t ∧ s = f − 1 ’t 11 23

Example: Product of Topologies What do we want : _ × _ : topo α → topo β → topo ( α × β ) cont π 1 (T 1 × T 2 ) T 1 cont π 2 (T 1 × T 2 ) T 2 cont ( π 1 ◦ f) T T 1 → cont ( π 2 ◦ f) T T 2 → cont f T (T 1 × T 2 ) Traditional definition : T 1 × T 2 = generate {s × t | is_open T 1 s ∧ is_open T 2 t} Generic definition (set, filter, topology, σ -algebra): T 1 × T 2 = comap π 1 T 1 ⊔ comap π 2 T 2 12 23

Example: Lattice Structure on Topologies Supremum as generating by the union: T 1 ⊔ T 2 = generate {s | is_open T 1 s ∨ is_open T 2 t} P roblem : Proof that it is a supremum. Define directly complete lattice over topologies: � � � s � is _ open T s � � T = � T ∈ T Problem : we might want different def eq for ⊤ , ⊓ , and � . Solution : Use Galois insertion (extending a Galois connection) to get the complete lattice structure 13 23

C onstructions N T N := ⊤ Indiscrete ⊥ Sum T α ⊕ T β := map ι 1 T α ⊓ map ι 2 T β def = { s | open ( ι − 1 1 [ s ]) ∧ open ( ι − 1 2 [ s ]) } Product T α × T β := comap π 1 T α ⊔ comap π 2 T β Pi ∏ i T α i := � i comap ( λω , ω i ) T α i Sigma Σ i T α i := � i map ( λ a , ( i , a )) T α i List (not done yet) list T α := lfp ( λ T , T () ⊕ T α × T ) list T α := gfp ( λ T , T () ⊕ T α × T ) 14 23

T actics and Commands

T heories

mathlib Basic (computable) data Type class hierarchies: Orders orders, lattices Algebraic (commutative) groups, rings, fields Spaces measurable, topological, uniform, metric Cardinals & ordinals Analysis: topology, measure theory, ... Linear algebra Group / ring / field theory Number theory 15 23

Cardinals and Ordinals Definition (Equivalence, Cardinals and Ordinals) structure α ≃ β := ( f : α → β ) ( g : β → α ) ( f _ g : f ◦ g = id ) ( g _ f : g ◦ f = id ) cardinal u Type u + 1 Type u / ≃ : := ordinal u Type u + 1 Well _ order u / ≃ ord : := Well-ordered, semiring, etc... Semiring structure of cardinal from ≃ constructions κ + κ = κ = κ ∗ κ (for κ ≥ ω ) Existence of inaccessible cardinals (i.e. in the next universe) Application: Dimension of subspaces 16 23

Number Theory p - adic numbers Q p by Rob Y. Lewis Cauchy construction of Q "in the other direction" Interesting result: Hensel’s lemma (see Rob’s talk) Quadratic Reciprocity by Chris Hughes ( ∼ 1300 lines patch) theorem quadratic_reciprocity (hp : prime p) (hq : prime q) (hpq : p � = q) (hp1 : p % 2 = 1) (hq1 : q % 2 = 1) : legendre p q hq * legendre q p hp = (-1) ^ (p/2 * q/2) 17 23

Analysis F ilter generalizes limits (derived from Isabelle/HOL) Topology nhds filter, open & closed & compact sets, interior, closure Uniformity complete, totally bounded, completion compact ↔ complete & totally bounded Metric instance of uniformities Norm only rudimentary (no deriviatives yet...) Measure Lebesgue measure etc... 18 23

Analysis: Constructing Reals Construct R using completion of Q For foundational reasons metric completion is not possible (alt: α → α → Q → Prop , c.f. Krebbers & Spitters) Formalize uniform spaces (using the filter library!) Use completion on the uniform space Q Is it worth it? Mario went back to Cauchy sequences... Anyway: R as order & topologically complete field 19 23

Measure Theory (Outer) Measures structure outer_measure ( α : Type*) := ( µ : set α → ennreal) ... structure measure ( α ) [measurable_space α ] extends outer_measure α := ... Outer measures provide natural totalization of measures Carathéodory’s extension theorem Lebesgue Measure Completion measurable space 20 23

P robability Theory def pmf ( α ) := { f : α → nnreal // is_sum f 1 } def pure (a : α ) : pmf α := � λ a’, if a’ = a then 1 else 0, is_sum_ite _ _ � def bind (p : pmf α ) (f : α → pmf β ) : pmf β := � λ b, ( ∑ a, p a * f a b), ... � prove rules of the Giry monad generality of ∑ helps! 21 23

Analysis: Infinite Sum Definition (Infinite sum) f i f i ∑ s → at _ top ∑ = lim i : ι i ∈ s Assuming: f : ι → α , [ topological _ add _ monoid α ] at _ top : filter ( finset ι ) finite sets approaching univ : set ι R , normed vector spaces, ennreal , Q , N , Z , ... ∑ b ∑ c f ( b , c ) = ∑ ( b , c ) f ( b , c ) — α regular ∑ n : N f n = lim i → ∞ ∑ i n = 0 f n 22 23

Linear Algebra Definition (Module) class module ( α : out_param (Type u)) ( β : Type v) [out_param (ring α )] [add_comm_group β ] extends semimodule α β C onstr.: Subspace, Linear maps, Quotient, Product, Tensor Product Dim.: dim ( β ) [vector_space α β ] : cardinal Laws: dom ( f ) s ∩ t ≃ ℓ s ⊕ t s ker ( f ) ≃ ℓ im ( f ) t R ⊕ M ≃ ℓ M M ⊕ N ≃ ℓ N ⊕ M M ⊕ ( N ⊕ L ) ≃ ℓ ( M ⊕ N ) ⊕ L 23 / 23