mathlib: Lean s mathematical library

Transcription

1 mathlib: Lean s mathematical library Johannes Hölzl EUTypes 2018

2 Introduction (classical) mathematical library for Lean with computable exceptions, e.g. N, Z, lists,...

3 Introduction (classical) mathematical library for Lean with computable exceptions, e.g. N, Z, lists,... Formerly distributed with Lean itself Leo wanted more flexibility

4 Introduction (classical) mathematical library for Lean with computable exceptions, e.g. N, Z, lists,... Formerly distributed with Lean itself Leo wanted more flexibility Some (current) topics: Basic Datatypes, Analysis, Linear Algebra, Set Theory,...

5 Lean

6 Lean DTT Quotient types (implies funext)

7 Lean DTT Quotient types (implies funext) Proof irrelevance is a definitional equality

8 Lean DTT Quotient types (implies funext) Proof irrelevance is a definitional equality Mark constants noncomputable, i.e. functions using choice: axiom choice : Π(α : Sort u), nonempty α α

9 Lean DTT Quotient types (implies funext) Proof irrelevance is a definitional equality Mark constants noncomputable, i.e. functions using choice: axiom choice : Π(α : Sort u), nonempty α α Non-commulative universes Prop : Type 0 : Type 1 :

10 Lean DTT Quotient types (implies funext) Proof irrelevance is a definitional equality Mark constants noncomputable, i.e. functions using choice: axiom choice : Π(α : Sort u), nonempty α α Non-commulative universes Prop : Type 0 : Type 1 : Basic inductives in the kernel

11 Lean DTT Quotient types (implies funext) Proof irrelevance is a definitional equality Mark constants noncomputable, i.e. functions using choice: axiom choice : Π(α : Sort u), nonempty α α Non-commulative universes Prop : Type 0 : Type 1 : Basic inductives in the kernel Mutual and nested inductives are constructed

12 Lean DTT Quotient types (implies funext) Proof irrelevance is a definitional equality Mark constants noncomputable, i.e. functions using choice: axiom choice : Π(α : Sort u), nonempty α α Non-commulative universes Prop : Type 0 : Type 1 : Basic inductives in the kernel Mutual and nested inductives are constructed No general fixpoint operator, no general match operator these are derived from recursors

13 Type classes in Lean Type classes are used to fill in implicit values: add : Π{α : Type}[i : has add α], α α α a + N nat.add a b

14 Type classes in Lean Type classes are used to fill in implicit values: add : Π{α : Type}[i : has add α], α α α a + N nat.add a b Instances can depend on other instances: ring.to group : Π(α : Type)[i : ring α] : group α

15 Type classes in Lean Type classes are used to fill in implicit values: add : Π{α : Type}[i : has add α], α α α a + N nat.add a b Instances can depend on other instances: ring.to group : Π(α : Type)[i : ring α] : group α Output parameters: has mem : Type out Type Type set.has mem : Πα, has mem (set α) α fset.has mem : Πα [i : decidable eq α], has mem (fset α) α

16 Type classes in Lean Type classes are used to fill in implicit values: add : Π{α : Type}[i : has add α], α α α a + N nat.add a b Instances can depend on other instances: ring.to group : Π(α : Type)[i : ring α] : group α Output parameters: has mem : Type out Type Type set.has mem : Πα, has mem (set α) α fset.has mem : Πα [i : decidable eq α], has mem (fset α) α Default values

17 Library

18 Library Basic (computable) data Type class hierarchies: Orders orders, lattices Algebraic (commutative) groups, rings, fields Spaces measurable, topological, uniform, metric Set theory (cardinals & ordinals) Analysis Linear algebra

19 Basic (computable) data Numbers: N, Z (as datatype, not quotient), Q, Fin n

20 Basic (computable) data Numbers: N, Z (as datatype, not quotient), Q, Fin n Lists, set α := α Prop

21 Basic (computable) data Numbers: N, Z (as datatype, not quotient), Q, Fin n Lists, set α := α Prop multiset α := list α/ perm

22 Basic (computable) data Numbers: N, Z (as datatype, not quotient), Q, Fin n Lists, set α := α Prop multiset α := list α/ perm finset α := {m : multiset α nodup m}

23 Basic (computable) data Numbers: N, Z (as datatype, not quotient), Q, Fin n Lists, set α := α Prop multiset α := list α/ perm finset α := {m : multiset α nodup m} Big operators for list, multiset and finset

24 Set theory: Cardinals and Ordinals (Mario Carneiro) Zorn s lemma, Schröder-Bernstein,...

25 Set theory: Cardinals and Ordinals (Mario Carneiro) Zorn s lemma, Schröder-Bernstein,... Isomorphism: structure α β := (f : α β)(g : β α)(f g : f g = id)(g f : g f = id)

26 Set theory: Cardinals and Ordinals (Mario Carneiro) Zorn s lemma, Schröder-Bernstein,... Isomorphism: structure α β := (f : α β)(g : β α)(f g : f g = id)(g f : g f = id) Cardinals & ordinals are well-order cardinal u : Type u+1 := Type u / nonempty ordinal u : Type u+1 := Well order u / nonempty ord

27 Set theory: Cardinals and Ordinals (Mario Carneiro) Zorn s lemma, Schröder-Bernstein,... Isomorphism: structure α β := (f : α β)(g : β α)(f g : f g = id)(g f : g f = id) Cardinals & ordinals are well-order cardinal u : Type u+1 := Type u / nonempty ordinal u : Type u+1 := Well order u / nonempty ord Semiring structure of cardinal is proved using constructions

28 Set theory: Cardinals and Ordinals (Mario Carneiro) Zorn s lemma, Schröder-Bernstein,... Isomorphism: structure α β := (f : α β)(g : β α)(f g : f g = id)(g f : g f = id) Cardinals & ordinals are well-order cardinal u : Type u+1 := Type u / nonempty ordinal u : Type u+1 := Well order u / nonempty ord Semiring structure of cardinal is proved using constructions κ + κ = κ = κ κ (for κ ω)

29 Set theory: Cardinals and Ordinals (Mario Carneiro) Zorn s lemma, Schröder-Bernstein,... Isomorphism: structure α β := (f : α β)(g : β α)(f g : f g = id)(g f : g f = id) Cardinals & ordinals are well-order cardinal u : Type u+1 := Type u / nonempty ordinal u : Type u+1 := Well order u / nonempty ord Semiring structure of cardinal is proved using constructions κ + κ = κ = κ κ (for κ ω) Existence of inaccessible cardinals (i.e. in the next universe)

30 Analysis Derived from Isabelle s analysis (Filters to generalize limits)

31 Analysis Derived from Isabelle s analysis (Filters to generalize limits) Topology: open; nhds filter, closed, compact, interior, closure

32 Analysis Derived from Isabelle s analysis (Filters to generalize limits) Topology: open; nhds filter, closed, compact, interior, closure Uniformity: complete, totally bounded (compact complete and totally bounded)

33 Analysis Derived from Isabelle s analysis (Filters to generalize limits) Topology: open; nhds filter, closed, compact, interior, closure Uniformity: complete, totally bounded (compact complete and totally bounded) Metric spaces are only rudimentary

34 Analysis Derived from Isabelle s analysis (Filters to generalize limits) Topology: open; nhds filter, closed, compact, interior, closure Uniformity: complete, totally bounded (compact complete and totally bounded) Metric spaces are only rudimentary Measurable spaces, Measures & Lebesgue measure

35 Analysis Derived from Isabelle s analysis (Filters to generalize limits) Topology: open; nhds filter, closed, compact, interior, closure Uniformity: complete, totally bounded (compact complete and totally bounded) Metric spaces are only rudimentary Measurable spaces, Measures & Lebesgue measure Infinite sum on topological monoids α: Σ : ι, (ι α) α

36 Analysis: Analytical Structures as Complete Lattices Complete lattices, map & comap as category theory light Filters, topological spaces, uniform spaces, and measurable spaces form a complete lattices per type complete lattice (topology α)

37 Analysis: Analytical Structures as Complete Lattices Complete lattices, map & comap as category theory light Filters, topological spaces, uniform spaces, and measurable spaces form a complete lattices per type complete lattice (topology α) (Co) induced structures allow for easy constructions: map : Π{αβ}, (α β) (topology α topology β) comap : Π{αβ}, (α β) (topology β topology α)

38 Analysis: Analytical Structures as Complete Lattices Complete lattices, map & comap as category theory light Filters, topological spaces, uniform spaces, and measurable spaces form a complete lattices per type complete lattice (topology α) (Co) induced structures allow for easy constructions: map : Π{αβ}, (α β) (topology α topology β) comap : Π{αβ}, (α β) (topology β topology α) Easy constructions: prod t 1 t 2 := comap π 1 t 1 comap π 2 t 2 subtype t s := comap (subtype.val s) t

39 Analysis: Analytical Structures as Complete Lattices Complete lattices, map & comap as category theory light Filters, topological spaces, uniform spaces, and measurable spaces form a complete lattices per type complete lattice (topology α) (Co) induced structures allow for easy constructions: map : Π{αβ}, (α β) (topology α topology β) comap : Π{αβ}, (α β) (topology β topology α) Easy constructions: prod t 1 t 2 := comap π 1 t 1 comap π 2 t 2 subtype t s := comap (subtype.val s) t Straight forward derivation of continuity rules

40 Analysis: Type Class Structure class metric (α : Type) :=... instance m2t (α : Type) [metric α] : topology α := {open s := x s, ɛ > 0, ball x ɛ s,...}

41 Analysis: Type Class Structure class metric (α : Type) :=... instance m2t (α : Type) [metric α] : topology α := {open s := x s, ɛ > 0, ball x ɛ s,...} Problem: m2t (m 1 m 2 ) (m2t m 1 ) (m2t m 2 )

42 Analysis: Type Class Structure class metric (α : Type) :=... instance m2t (α : Type) [metric α] : topology α := {open s := x s, ɛ > 0, ball x ɛ s,...} Problem: m2t (m 1 m 2 ) (m2t m 1 ) (m2t m 2 ) class metric (α : Type) extends topology α :=... (open iff : s, open s x s, ɛ > 0, ball x ɛ s)

43 Analysis: Type Class Structure class metric (α : Type) :=... instance m2t (α : Type) [metric α] : topology α := {open s := x s, ɛ > 0, ball x ɛ s,...} Problem: m2t (m 1 m 2 ) (m2t m 1 ) (m2t m 2 ) class metric (α : Type) extends topology α :=... (open iff : s, open s x s, ɛ > 0, ball x ɛ s) Default values give a value for the topology when defining metric

44 Analysis: Constructing Reals Construct R using completion of Q For foundational reasons metric completion is not possible

45 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)

46 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!)

47 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

48 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?

49 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 wants to go back to Cauchy sequences...

50 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 wants to go back to Cauchy sequences... Anyway: R as order & topologically complete field

51 Linear Algebra class module (α : out Type u ) (β : Type v ) [out ring α] :=...

52 Linear Algebra class module (α : out Type u ) (β : Type v ) [out ring α] :=... type class mechanism looks for module β

53 Linear Algebra class module (α : out Type u ) (β : Type v ) [out ring α] :=... type class mechanism looks for module only one canoncial module per type β

54 Linear Algebra class module (α : out Type u ) (β : Type v ) [out ring α] :=... type class mechanism looks for module only one canoncial module per type usually α is fixed per theory anyway β

55 Linear Algebra class module (α : out Type u ) (β : Type v ) [out ring α] :=... type class mechanism looks for module only one canoncial module per type usually α is fixed per theory anyway Problem: (multivariate) polynomials β

56 Linear Algebra class module (α : out Type u ) (β : Type v ) [out ring α] :=... type class mechanism looks for module only one canoncial module per type usually α is fixed per theory anyway Problem: (multivariate) polynomials β Constructions: Subspace, Linear maps, Quotient, Product

57 Linear Algebra class module (α : out Type u ) (β : Type v ) [out ring α] :=... type class mechanism looks for module only one canoncial module per type usually α is fixed per theory anyway Problem: (multivariate) polynomials β Constructions: Subspace, Linear maps, Quotient, Product Example Isomorphism laws: dom(f ) ker(f ) l im(f ) s s t l s t t

58 Discussion

59 Problems with Type Classes group monoid semigroup

60 Problems with Type Classes group comm group monoid comm monoid semigroup comm semigroup

61 Problems with Type Classes add group add comm group group comm group add monoid add comm monoid monoid comm monoid add semigroup add comm semigroup semigroup comm semigroup

62 Problems with Type Classes add group add comm group group comm group add monoid add comm monoid monoid comm monoid add semigroup add comm semigroup semigroup comm semigroup Currently a automated copy from group to add group instead: [is group( )(/)( 1 )1] and [is group(+)( )( )0] Mixin type classes replace comm monoid,... by [is commutative ( )]

63 Problem with Universes class functor (M : Type u Type v) := (map : (α β : Type u), (α β) M α M β) (map comp : (α β γ : Type u) f g h, map f map g = map (f g)) (map id : α, map id = id)

64 Problem with Universes Problematic u class functor (M : Type u Type v) := (map : (α β : Type u), (α β) M α M β) (map comp : (α β γ : Type u) f g h, map f map g = map (f g)) (map id : α, map id = id)

65 Problem with Universes Problematic u class functor (M : Type u Type v) := (map : (α β : Type u), (α β) M α M β) (map comp : (α β γ : Type u) f g h, map f map g = map (f g)) (map id : α, map id = id) If we only work with functor (topology α) our library is too limited, e.g. topology.map allows mapping between different universes.

66 Maintenance Currently maintained by Mario Carneiro, me, and Jeremy Avigad Contributors: Andrew Zipperer, Floris van Doorn, Haitao Zhang, Jeremy Avigad, Johannes Hölzl, Kenny Lau, Kevin Buzzard, Leonardo de Moura, Mario Carneiro, Minchao Wu, Nathaniel Thomas, Parikshit Khanna, Robert Y. Lewis, Simon Hudon Currently lines of Lean code

67 mathlib A (classical) mathematical library for Lean