![20
Sized Types Contributions Implementation
Sized Typing Pipeline
Definition mynat [u] : Set ≔ natu
.
Fixpoint div [v, w, …] (n : mynatv
) (m : mynatw
) : mynatv
≔ …
Coq kernel](https://image.slidesharecdn.com/issizedtypingforcoqpractical-2-230925175151-f5db7ed9/85/Is-Sized-Typing-for-Coq-Practical-20-320.jpg)
![🆕 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](https://image.slidesharecdn.com/issizedtypingforcoqpractical-2-230925175151-f5db7ed9/85/Is-Sized-Typing-for-Coq-Practical-27-320.jpg)
![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](https://image.slidesharecdn.com/issizedtypingforcoqpractical-2-230925175151-f5db7ed9/85/Is-Sized-Typing-for-Coq-Practical-30-320.jpg)
![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](https://image.slidesharecdn.com/issizedtypingforcoqpractical-2-230925175151-f5db7ed9/85/Is-Sized-Typing-for-Coq-Practical-31-320.jpg)
![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](https://image.slidesharecdn.com/issizedtypingforcoqpractical-2-230925175151-f5db7ed9/85/Is-Sized-Typing-for-Coq-Practical-36-320.jpg)
Is Sized Typing for Coq Practical?
- 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