Slides for JFP talk at ICFP 2023: https://icfp23.sigplan.org/details/icfp-2023-papers/39

1. Journal of Functional Programming, 33(e1), January 2023
Is Sized Typing for Coq Practical?
Jonathan Chan
University of British Columbia,
University of Pennsylvania
Yufeng (Michael) Li
University of Waterloo,
University of Cambridge
William J. Bowman
University of British Columbia

2. 1. Sized Types
2. Contributions
3. Implementation
2

3. Termination Checking: Guardedness
Fixpoint minus n m : nat :=
match n, m with
| S n′, S m′ => minus n′ m′
| _, _ => O
end.
3
Sized Types Contributions Implementation
✅

4. Termination Checking: Guardedness
4
Sized Types Contributions Implementation
Fixpoint div n m : nat :=
match n with
| S n′ => S (div minus n′ m m)
| O => O
end.
❌
Fixpoint minus n m : nat :=
match n, m with
| S n′, S m′ => minus n′ m′
| _, _ => O
end.

5. Termination Checking: Guardedness
5
Sized Types Contributions Implementation
Fixpoint div n m : nat :=
match n with
| S n′ => S (div minus n′ m m)
| O => O
end.
❌
Fixpoint minus n m : nat :=
match n, m with
| S n′, S m′ => minus n′ m′
| _, _ => O n
end.
❌
✅

6. Termination Checking: Sized Typing
6
Sized Types Contributions Implementation
Γ ⊢ n : nats+1
Γ ⊢ e1
: P O
Γ ⊢ n : nats
Γ, m : nats
⊢ e2
: P (S m)
───────────── ─────────────── ─────────────────────────
Γ ⊢ O : nats+1
Γ ⊢ S n : nats+1
Γ ⊢ match n with .
| O => e1
| S m => e2
end : P n
s ⩴ v | s+1 | ∞

7. Termination Checking: Sized Typing
7
Sized Types Contributions Implementation
Γ ⊢ n : nats+1
Γ ⊢ e1
: P O
Γ ⊢ n : nats
Γ, m : nats
⊢ e2
: P (S m)
───────────── ─────────────── ─────────────────────────
Γ ⊢ O : nats+1
Γ ⊢ S n : nats+1
Γ ⊢ match n with .
| O => e1
| S m => e2
end : P n

8. Termination Checking: Sized Typing
8
Sized Types Contributions Implementation
Fixpoint minus : natv
-> nat -> natv
.
Fixpoint div (n : natv+1
) (m : nat) : natv+1
:=
match n with
| S n′ => S (div (minus n′ m) m)
| O => O
end.
natv
natv

9. Past Work
• dependent types
• size inference algorithm
• nested (co)inductives
• universes
9
Sized Types Contributions Implementation
Hughes & Pareto & Sabry 1996
𝒞𝒞ℛ (Giménez 1998)
Amadio & Coupet-Grimal 1998
λ^ (Barthe et al. 2002; Frade 2004)
Λμ+
(Abel 2004)
λfixμν
(Abel 2003)
F^ (Barthe et al. 2005)
F^ω (Abel 2006)
F^× (Barthe et al. 2008)
MiniAgda (Abel 2010, 2012)
CIC^ (Barthe et al. 2006)
CIC^- (Grégoire & Sacchini 2010;
Sacchini 2011)
CCω
̂ (Sacchini 2013)
CIC^l (Sacchini 2014)
CIC^⊑ (Sacchini 2015)
Fcop
ω (Abel & Pientka 2016)
Abel 2017
─ ─ ─

10. Contributions
• dependent types
• size inference algorithm
• nested (co)inductives
• universes
10
Sized Types Contributions Implementation
CIC^ (Barthe et al. 2006)
CIC^- (Grégoire & Sacchini 2010;
Sacchini 2011)
CCω
̂ (Sacchini 2013)
CIC^*
(2019 – 2023)
─ ─ ─

11. Contributions
• dependent types
• annotation inference alg.
• nested (co)inductives
• universes
11
Sized Types Contributions Implementation
CIC^ (Barthe et al. 2006)
CIC^- (Grégoire & Sacchini 2010;
Sacchini 2011)
CCω
̂ (Sacchini 2013)
CIC^*
(2019 – 2023)
─ ─ ─

12. Contributions
• dependent types
• annotation inference alg.
• nested (co)inductives
• universes
12
Sized Types Contributions Implementation
CIC^ (Barthe et al. 2006)
CIC^- (Grégoire & Sacchini 2010;
Sacchini 2011)
CCω
̂ (Sacchini 2013)
CIC^*
(2019 – 2023)
─ ─ ─

13. Contributions
• dependent types
• annotation inference alg.
• nested (co)inductives
• cumulative universes
• global + local definitions
13
Sized Types Contributions Implementation
CIC^ (Barthe et al. 2006)
CIC^- (Grégoire & Sacchini 2010;
Sacchini 2011)
CCω
̂ (Sacchini 2013)
CIC^*
(2019 – 2023)
─ ─ ─

14. Contributions
• implementation in Coq fork
• dependent types
• annotation inference alg.
• nested (co)inductives
• cumulative universes
• global + local definitions
14
Sized Types Contributions Implementation
CIC^ (Barthe et al. 2006)
CIC^- (Grégoire & Sacchini 2010;
Sacchini 2011)
CCω
̂ (Sacchini 2013)
CIC^*
(2019 – 2023)
─ ─ ─

15. Goal: backward-compatible sized typing for Coq for
expressive, modular, efficient termination checking
15
Sized Types Contributions Implementation

16. Goal: backward-compatible sized typing for Coq for
expressive, modular, efficient termination checking
16
Sized Types Contributions Implementation
NOPE

17. 17
Sized Types Contributions Implementation
Sized Typing Pipeline
CIC CIC^* + 𝒞 CIC^*
solve &
substitute
type checking
size inference
Coq kernel

18. 18
Sized Types Contributions Implementation
Sized Typing Pipeline
Coq kernel
𝒞; Γ ⊢ e : τ ⇝ 𝒞′; e′ : τ′ ⇝ e″ : τ″

19. 19
Sized Types Contributions Implementation
Sized Typing Pipeline
Coq kernel
𝒞; Γ ⊢ e : τ ⇝ 𝒞′; e′ : τ′ ⇝ e″ : τ″

20. 20
Sized Types Contributions Implementation
Sized Typing Pipeline
Definition mynat [u] : Set ≔ natu
.
Fixpoint div [v, w, …] (n : mynatv
) (m : mynatw
) : mynatv
≔ …
Coq kernel

21. Size Inference: Fixpoints
𝒞; Γ, f : τv
⊢ … ⇝ 𝒞′; e : τv+1
SAT(𝒞′ ∪ {v ≠ ∞})
───────────────────────────────────────────────────
𝒞; Γ ⊢ … ⇝ 𝒞′; fix f : τ*
≔ e : τv
21
Sized Types Contributions
𝒞; Γ ⊢ e : τ ⇝ 𝒞′; e′ : τ′
Inference Substitution Performance

22. Size Inference: Fixpoints
𝒞; Γ, f : τv
⊢ … ⇝ 𝒞′; e : τv+1
SAT(𝒞′ ∪ {v ≠ ∞})
───────────────────────────────────────────────────
𝒞; Γ ⊢ … ⇝ 𝒞′; fix f : τ*
≔ e : τv
22
Sized Types Contributions
𝒞; Γ ⊢ e : τ ⇝ 𝒞′; e′ : τ′
Inference Substitution Performance
• nat*
→ nat → nat
• nat → nat*
→ nat

23. Size Inference: Fixpoints
𝒞; Γ, f : τv
⊢ … ⇝ 𝒞′; e : τv+1
SAT(𝒞′ ∪ {v ≠ ∞})
───────────────────────────────────────────────────
𝒞; Γ ⊢ … ⇝ 𝒞′; fix f : τ*
≔ e : τv
23
Sized Types Contributions
𝒞; Γ ⊢ e : τ ⇝ 𝒞′; e′ : τ′
Inference Substitution Performance
• nat*
→ nat → nat*
• nat*
→ nat → nat
• nat → nat*
→ nat*
• nat → nat*
→ nat

24. Size Inference: Cumulativity
𝒞; Γ ⊢ … ⇝ 𝒞′; e : τ′ τ′ ≼ τ ⇝ 𝒞″
─────────────────────────────────────
𝒞; Γ ⊢ … ⇝ 𝒞′ ∪ 𝒞″; e : τ
24
Sized Types Contributions
𝒞; Γ ⊢ e : τ ⇝ 𝒞′; e′ : τ′
Inference Substitution Performance

25. Size Inference: Constraints
nats₁
≼ nats₂
⇝ {s₁ ⊑ s₂}
conats₁
≼ conats₂
⇝ {s₂ ⊑ s₁}
25
Sized Types Contributions
τ1
≼ τ2
⇝ 𝒞
𝒞 ⩴ {v1
+ n1
⊑ v2
+ n2
, …}
Inference Substitution Performance

26. Size Inference: Constraint Satisfiability
nats₁
≼ nats₂
⇝ {s₁ ⊑ s₂}
conats₁
≼ conats₂
⇝ {s₂ ⊑ s₁}
─────── ──────
s ⊑ s+1 s ⊑ ∞
26
Sized Types Contributions
τ1
≼ τ2
⇝ 𝒞
Inference Substitution Performance

27. 🆕 Constraint Solving
ρ = SAT(𝒞)
───────────────────
𝒞; e : τ ⇝ ρe : ρτ
27
Sized Types Contributions
𝒞′; e′ : τ′ ⇝ e″ : τ″
ρ ⩴ [v1
↦ s1
, …]
Inference Substitution Performance

28. 🆕 Constraint Solving
ρ = SAT(𝒞)
───────────────────
𝒞; e : τ ⇝ ρe : ρτ
28
Sized Types Contributions
negative cycle
detection
e.g. Bellman–Ford
O(|𝒱||𝒞|)
𝒞′; e′ : τ′ ⇝ e″ : τ″
Inference Substitution Performance

29. Artificial Example: Exponentially Many `nat`s
Time Definition nats1 :=
(nat, nat, nat, nat, nat, nat, nat, nat).
Time Definition nats2 :=
(nats1, nats1, nats1, nats1).
...
Time Definition nats6 :=
(nats5, nats5, nats5, nats5).
29
Sized Types Contributions Inference Substitution Performance

30. Artificial Example: Exponentially Many `nat`s
Definition nats1 [v1, ..., v21] :
Set *v1
Set *v2
Set *v3
Set *v4
Set *v5
Set *v6
Set *v7
Set ≔
(natv8
, natv9
, natv10
, natv11
, natv12
, natv13
, natv14
, natv15
)v16,v17,v18,v19,v20,v21
.
Definition nats2 [...] ≔ (nats1v1,…,v21
, nats1v22,…,v42
, nats1…
, nats1…
).
...
𝒞nats1
= {v16 ⊑ v1, v17 ⊑ v2, …}
30
Sized Types Contributions Inference Substitution Performance

31. Artificial Example: Exponentially Many `nat`s
Definition nats1 [v1, ..., v21] :
Set *v1
Set *v2
Set *v3
Set *v4
Set *v5
Set *v6
Set *v7
Set ≔
(natv8
, natv9
, natv10
, natv11
, natv12
, natv13
, natv14
, natv15
)v16,v17,v18,v19,v20,v21
.
Definition nats2 [...] ≔ (nats1v1,…,v21
, nats1v22,…,v42
, nats1…
, nats1…
).
...
𝒞nats1
= {v16 ⊑ v1, v17 ⊑ v2, …}
31
Sized Types Contributions Inference Substitution Performance

32. Artificial Example: Exponentially Many `nat`s
Time Definition nats1 :=
(nat, nat, nat, nat,
nat, nat, nat, nat).
Time Definition nats2 :=
(nats1, nats1, nats1, nats1).
...
Time Definition nats6 :=
(nats5, nats5, nats5, nats5).
32
Sized Types Contributions
Definition |𝒱| Time (s)
nats1 21 0.004
nats2 93 0.020
nats3 381 0.177
nats4 1533 2.299
nats5 6141 35.385
nats6 24573 > 120
Inference Substitution Performance

33. Real Example: MSets/MSetList.v
Mean over five trials
33
Sized Types Contributions
Unsized compilation (s) 15.122 ± 0.073
Sized compilation (s) 83.660 ± 0.286
Slowdown 5.5×
SAT ops only (s) 64.600 ± 0.437
SAT ops only (%) 77.2%
Inference Substitution Performance

34. Real Example: MSets/MSetList.v
34
Sized Types Contributions
log(count)
log distribution of |𝒱| × |𝒞| during SAT operations
|𝒱| × |𝒞|
Inference Substitution Performance
~4K vs. × ~250 cs.

35. size inference
35
backward-compatible sized typing
definitions
constraint checking/solving
poor performance
Sized Types Contributions Implementation

36. Future Work?: Manual Size Annotation + Size Inference
36
Fixpoint minus [v] : natv
→ nat∞
→ natv
.
Fixpoint div [v] (n : natv
) (m : nat∞
) : natv
≔
match n with
| S n′ ⇒ S (div [?w] (minus [?u] n′ m) m)
| O ⇒ O
End.
𝒞div
= {?u = ?w, ?w+1 ⊑ v}
Sized Types Contributions Implementation

37. Thank you!
● Full spec of size
inference algorithm
& soundness wrt typing
● Metatheoretical results
& attempt at
set-theoretic model
● More detailed
performance profiling
& statistics
+ Closed draft PR:
https://github.com/coq/
coq/pull/12426
37
Check out the paper for:

38. 38
bonus content

39. Concrete sized naturals
O : natv+1
S O : natv+2
S (S O) : natv+3
O : natv+2
S O : natv+3
S (S O) : natv+4
39

40. Real Example: setoid_ring/Field_theory.v
40
Mean over two trials (August 2023)
Unsized compilation (s) 17.815 ± 0.545
Sized compilation (s) 106.87 ± 1.94
Slowdown 6.0×
SAT ops only (s) 84.70
SAT ops only (%) 79.3% 17755 vs. × 14057 cs. ≈ 250M

41. Artificial Example: Universe Polymorphism
Set Printing Universes.
Set Universe Polymorphism.
Time Definition T1 : Type := Type -> Type
-> Type -> Type -> Type -> Type. Print T1.
Time Definition T2 : Type :=
T1 -> T1 -> T1 -> T1 -> T1 -> T1. Print T2.
Time Definition T3 : Type :=
T2 -> T2 -> T2 -> T2 -> T2 -> T2. Print T3.
Time Definition T4 : Type :=
T3 -> T3 -> T3 -> T3 -> T3 -> T3. Print T4.
Time Definition T5 : Type :=
T4 -> T4 -> T4 -> T4 -> T4 -> T4. Print T5.
Time Definition T6 : Type :=
T5 -> T5 -> T5 -> T5 -> T5 -> T5. Print T6.
Time Definition T7 : Type :=
T6 -> T6 -> T6 -> T6 -> T6 -> T6. Print T7.
41
Definition #u Time (s)
T1 7 ~ 0
T2 43 0.002
T3 259 0.026
T4 1555 0.057
T5 9331 0.374
T6 55987 3.300
T7 335921 18.170