Vectors are records, too!

Transcription

1 Vectors are records, too! Jesper Cockx 1 Gaëtan Gilbert 2 Nicolas Tabareau 2 Matthieu Sozeau 2 1 Gothenburg University, Sweden 2 INRIA, France 21 June 2018

2 types most popular example 1 data V (A : Set) : (n : N) Set where nil : V A zero cons : (m : N)(x : A)(xs : V A m) V A (suc m) 1 Disclaimer: I did not actually count all examples since / 15

3 types most popular example 1 V : (A : Set)(n : N) Set V A zero = V A (suc n) = A V A n 1 Disclaimer: I did not actually count all examples since / 15

4 types most popular example 1 data V (A : Set) : (n : N) Set where nil : V A zero cons : (m : N)(x : A)(xs : V A m) V A (suc m) vs. V : (A : Set)(n : N) Set V A zero = V A (suc n) = A V A n 1 Disclaimer: I did not actually count all examples since / 15

5 Presenting... A common representation of indexed datatypes and recursive types as case-splitting datatypes. An elaboration algorithm to automatically transform an indexed datatype into a case-splitting datatype. 2 / 15

6 Inductive types vs. recursive types Case-splitting datatypes Elaborating indexed datatypes

7 Inductive types vs. recursive types Case-splitting datatypes Elaborating indexed datatypes

8 Inductive type Recursive type 3 / 15

9 Inductive type I Intuitive notation Recursive type data V (A : Set) : (n : N) Set where nil : V A zero cons : (m : N)(x : A)(xs : V A m) V A (suc m) 3 / 15

10 Inductive type I Intuitive notation Recursive type I Eta equality x : V A zero x tt x : V A (suc m) x (x.π 1, x.π 2 ) 3 / 15

11 Inductive type II Intuitive notation Pattern matching Recursive type I Eta equality tail : (m : N)(xs : V A (suc m)) V A m tail m (cons m x xs) = xs 3 / 15

12 Inductive type II Intuitive notation Pattern matching Recursive type II Eta equality Forcing & detagging for free cons : (m : Nat)(x : A)(xs : V A m) V A (suc m) cons m x xs = (x, xs) 3 / 15

13 Inductive type III Intuitive notation Pattern matching Structural recursion Recursive type II Eta equality Forcing & detagging for free 3 / 15

14 Inductive type III Intuitive notation Pattern matching Structural recursion Recursive type III Eta equality Forcing & detagging for free Large indices : N N Prop zero n = (suc m) zero = (suc m) (suc n) = m n 3 / 15

15 Inductive type IIII Intuitive notation Pattern matching Structural recursion Non-indexed / non-stratified types Recursive type III Eta equality Forcing & detagging for free Large indices N = N is not a valid definition! 3 / 15

16 Inductive type IIII Intuitive notation Pattern matching Structural recursion Non-indexed / non-stratified types Recursive type IIII Eta equality Forcing & detagging for free Large indices Non-positive types data U : Set where : U U U = : U Set =(t 1 t 2 ) = = t 1 = t 2 3 / 15

17 Inductive types vs. recursive types Case-splitting datatypes Elaborating indexed datatypes

18 Indexed datatype data V A : N Set where... Recursive type V A zero =... V A (suc m) =... 4 / 15

19 Indexed datatype data V A : N Set where... Case-splitting datatype V A n = case n {...} Recursive type V A zero =... V A (suc m) =... 4 / 15

20 Indexed datatype Case-splitting datatype Recursive type data V A : N Set where... V A n = case n {...} V A zero =... V A (suc m) =... 4 / 15

21 Indexed datatype Case-splitting datatype Recursive type data V A : N Set where... V A n = case n {...} V A zero =... V A (suc m) =... 4 / 15

22 General syntax for case-splitting datatypes Q ::= c c k k case x c 1 ˆ 1 τ 1 Q 1. c n ˆ n τ n Q n 5 / 15

23 Case tree for V A n case n zero nil suc m cons (x : A)(xs : V A m) 6 / 15

24 Case tree for m n case m zero lz suc m case n zero suc n ls (p : m n ) 7 / 15

25 From case tree to a datatype: Ignore case splits; gather all constructors in a flat list. From case tree to a recursive definition: Translate case splits with tools from eliminating dependent pattern matching. 8 / 15

26 Inductive types vs. recursive types Case-splitting datatypes Elaborating indexed datatypes

27 Problem: We don t want to write case trees, we want to write datatypes! 9 / 15

28 Problem: We don t want to write case trees, we want to write datatypes! Solution: Elaborate datatypes to case trees automatically. 9 / 15

29 State of elaborating a datatype c 1 1 [Φ 1 ]. c k k [Φ k ] is outer telescope of datatype indices c 1,..., c k are the constructor names i is inner telescope of arguments of c i Φ i is a set of constraints {v j /? p j } 10 / 15

30 Initial elaboration state (A : Set)(n : N) { nil [zero /? n] } cons (m : N)(x : A)(xs : V A m) [suc m /? n] 11 / 15

31 Elaboration step: case split on index (A : Set)(n : N) { nil [zero /? n] } cons (m : N)(x : A)(xs : V A m) [suc m /? n] (A : Set) { nil [zero /? zero] } (A : Set)(n : N) { cons (m : N)(x : A)(xs : V A m) [suc m /? suc n ] } 11 / 15

32 Elaboration step: solve constraint (A : Set)(n : N) { cons (m : N)(x : A)(xs : V A m) [suc m /? suc n ] } (A : Set)(n : N) { cons (x : A)(xs : V A n ) } 12 / 15

33 Elaboration step: finish splitting (A : Set)(n : N) { cons (m : N)(x : A)(xs : V A m) [suc m /? suc n ] } (A : Set)(n : N) { cons (x : A)(xs : V A n ) } cons (x : A)(xs : V A n ) 12 / 15

34 Elaboration step: introduce equality proof (A B : Set)(f : A B)(y : B) { image (x : A) [f x /? y] } (A B : Set)(f : A B)(y : B) { image (x : A) (e : f x B y) } 13 / 15

35 Ongoing & future work Implement translation in Coq (WIP) Generate constructors & eliminator Generate case trees for Agda datatypes User syntax to control splitting? 14 / 15

36 Conclusion Datatypes have long been denied features of record types such as η-equality. 15 / 15

37 Conclusion Datatypes have long been denied features of record types such as η-equality. We can automatically transform a datatype into an equivalent definition with η-laws. 15 / 15

38 Conclusion Datatypes have long been denied features of record types such as η-equality. We can automatically transform a datatype into an equivalent definition with η-laws. Now you can both have the cake and eat it: vectors are records, too! 15 / 15