Sharing a Library between Proof Assistants: Reaching out the HOL - - PowerPoint PPT Presentation

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Sharing a Library between Proof Assistants: Reaching out the HOL - - PowerPoint PPT Presentation

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Sharing a Library between Proof Assistants: Reaching out the HOL Family Franc ois Thir e July 7, 2018 LSV, CNRS, Inria, ENS Paris-Saclay 1 Introduction Beluga PVS Automath Coq Agda RedPRL Isabelle/HOL Twelf Abella Matita

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SLIDE 1

Sharing a Library between Proof Assistants: Reaching out the HOL Family

Franc ¸ois Thir´ e July 7, 2018

LSV, CNRS, Inria, ENS Paris-Saclay 1

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SLIDE 2

Introduction

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SLIDE 3

Dedukti Coq Agda RedPRL Isabelle/HOL Matita Twelf Automath PVS Beluga Abella

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SLIDE 4

Dedukti Coq Agda RedPRL Isabelle/HOL Matita Twelf Automath PVS Beluga Abella

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SLIDE 5

Dedukti Coq Agda RedPRL Isabelle/HOL Matita Twelf Automath PVS Beluga Abella

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SLIDE 6

Dedukti

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SLIDE 7

Dedukti Matita OpenTheory

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SLIDE 8

OpenTheory HOL-Light HOL4 Isabelle/HOL HOL Zero

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SLIDE 9

Dedukti D[OT] D[Matita] OpenTheory Matita

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SLIDE 10

Dedukti D[STT∀βδ] D[CiCω] OpenTheory Matita

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SLIDE 11

Dedukti D[STT∀βδ] D[CiCω] OpenTheory Matita Coq

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SLIDE 12

Dedukti D[STT∀βδ] D[CiCω] OpenTheory Matita Coq Lean ? PVS

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SLIDE 13

STT∀βδ

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A real implementation of STT∀βδ?

Dedukti D[STT∀βδ] STT∀βδ

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SLIDE 15

A real implementation of STT∀βδ?

Dedukti D[STT∀βδ] STT∀βδ

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A real implementation of STT∀βδ?

Dedukti D[STT∀βδ] STT∀βδ In this talk, Dedukti is abstract! The encoding is shallow

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SLIDE 17

STT

Types A, B :≡ ι | o | A → B Terms t, u :≡ x | λxA. t | t u | ∀xA. t | t ⇒ u

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STT

Types A, B :≡ ι | o | A → B Terms t, u :≡ x | λxA. t | t u | ∀xA. t | t ⇒ u

C ⊢ t : o C, t ⊢ t assume C ⊢ t C ⊢ t ⇒ u C ⊢ u

⇒E

C, t ⊢ u C ⊢ t ⇒ u ⇒I C ⊢ ∀xA. t C ⊢ u : A C ⊢ t[x := u]

∀E

C, x : A ⊢ t x ∈ C C ⊢ ∀xA. t

∀I

  • Fig. 1: Proof system

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STTβδ

Types A, B :≡ ι | o | A → B Terms t, u :≡ x | λxA. t | t u | ∀xA. t | t ⇒ u

C ⊢ t : o C, t ⊢ t assume C ⊢ t t ≡βδ t′ C ⊢ t′

conv

C ⊢ t C ⊢ t ⇒ u C ⊢ u

⇒E

C, t ⊢ u C ⊢ t ⇒ u ⇒I C ⊢ ∀xA. t C ⊢ u : A C ⊢ t[x := u]

∀E

C, x : A ⊢ t x ∈ C C ⊢ ∀xA. t

∀I

  • Fig. 1: Proof system

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STT∀βδ is an extension of STT

STT∀βδ = STTβδ + prenex polymorphism

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STT∀βδ is an extension of STT

monotypes A, B :≡

  • | A → B | X | p A1 . . . An

polytypes T :≡ ∀KX. T | A

  • nat
  • ∀KX. list X
  • list nat
  • ∀KX. X→X→ o

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STT∀βδ is an extension of STT

monotypes A, B :≡

  • | A → B | X | p A1 . . . An

polytypes T :≡ ∀KX. T | A monoterms t,u :≡ ... | c A1 . . . An | ΛX. t polyterms τ :≡ A

  • X. τ | t
  • 0 : nat
  • ΛX. λxX. λyX. ∀PX→o. P x ⇒ P y : ∀KX. X → X → o

(eq)

  • A
  • X. ∀aX. eq X a a

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STT∀βδ is an extension of STT

monotypes A, B :≡

  • | A → B | X | p A1 . . . An

polytypes T :≡ ∀KX. T | A monoterms t,u :≡ ... | c A1 . . . An | ΛX. t polyterms τ :≡ A

  • X. τ | t

. . . C ⊢ A

  • X. τ

C ⊢ A wf C ⊢ τ[X := A]

A

E

C, X ⊢ τ C ⊢ A

  • X. τ

A

I

  • Fig. 2: Rules for STT∀βδ

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A reflexivity proof

eq; ∅; ∅ ⊢ A

  • X. ∀aX. eq X a a

A

I

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A reflexivity proof

eq; X; ∅ ⊢ ∀aX. eq X a a

∀I

eq; ∅; ∅ ⊢ A

  • X. ∀aX. eq X a a

A

I

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A reflexivity proof

eq; X, a : X; ∅ ⊢ eq X a a

conv

eq; X; ∅ ⊢ ∀aX. eq X a a

∀I

eq; ∅; ∅ ⊢ A

  • X. ∀aX. eq X a a

A

I

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A reflexivity proof

eq; X, a : X; ∅ ⊢ P a ⇒ P a

⇒I

eq; X, a : X; ∅ ⊢ eq X a a

conv

eq; X; ∅ ⊢ ∀aX. eq X a a

∀I

eq; ∅; ∅ ⊢ A

  • X. ∀aX. eq X a a

A

I

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A reflexivity proof

eq; X, a : X; P a ⊢ P a

assume

eq; X, a : X; ∅ ⊢ P a ⇒ P a

⇒I

eq; X, a : X; ∅ ⊢ eq X a a

conv

eq; X; ∅ ⊢ ∀aX. eq X a a

∀I

eq; ∅; ∅ ⊢ A

  • X. ∀aX. eq X a a

A

I

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STT∀βδ as a PTS

Γ ⊢ A : s1 Γ, x : A ⊢ B : s2 (s1, s2, s3) ∈ R Γ ⊢ (x : A) → B : s3

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STT∀βδ as a PTS

S, A = Prop : Type : Kind ∀K ∀ ⇒ → A (Type, Kind, Kind) (Type, Prop, Prop) (Prop, Prop, Prop) (Type, Type, Type) (Kind, Prop, Prop) Type ≺ Kind (subtyping)

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OpenTheory

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Dedukti D[STT∀βδ] OpenTheory Coq

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OpenTheory vs STT∀βδ

Terms and types are almost the same! Three main differences: In STT∀βδ:

  • β and δ extensional
  • ∀ and ⇒ connectives
  • ∀K is explicit

In OpenTheory:

  • β and δ intensional
  • = connective
  • ∀K is implicit

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OpenTheory vs STT∀βδ

In STT∀βδ:

  • β and δ extensional
  • ∀ and ⇒ connectives
  • ∀K is explicit

In OpenTheory:

  • β and δ intensional
  • = connective
  • ∀K is implicit

hard easy easy

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Why is it hard?

C ⊢ t t≡βδt′ C ⊢ t′

conv

  • ≡βδ is the one of Dedukti
  • How to annotate proofs? Reduce the term step by step.
  • β of STT∀βδ vs administrative β
  • Don’t compute the normal form everytime!

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Coq

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Dedukti D[STT∀βδ] OpenTheory Coq

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Trivial: STT∀βδ is a subsystem of Coq !

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DEMO

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Arithmetic library

Dedukti[STT] OpenTheory Coq Matita Lean PVS size (mb) 1.5 41 0.6 0.6 0.6 9 translation time (s)

  • 18

3 3 3 3 checking time (s) 0.1 13 6 2 1 ∼300 14

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Arithmetic library

Dedukti[STT] OpenTheory Coq Matita Lean PVS size (mb) 1.5 41 0.6 0.6 0.6 9 translation time (s)

  • 18

3 3 3 3 checking time (s) 0.1 13 6 2 1 ∼300

  • Theorems: 340 (Commutativity of addition, Fermat’s little

theorem)

  • Parameters: 46 (nat, bool, ...)
  • Axiom: 71 (equalities generated from recursive definitions,...)
  • Definitions: 34 (le,primes,...)

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Concept Alignement

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Fermat’s little theorem

Theorem congruent_exp_pred_SO : forall p a : Nat, prime p -> Not (divides p a) -> congruent (exp a (pred p)) (S O) p.

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Fermat’s little theorem

Theorem congruent_exp_pred_SO : forall p a : Nat, prime p -> Not (divides p a) -> congruent (exp a (pred p)) (S O) p.

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Fermat’s little theorem

Theorem congruent_exp_pred_SO : forall p a : Nat, prime p -> Not (divides p a) -> congruent (exp a (pred p)) (S O) p. Parameter exp : Nat -> Nat -> Nat. Axiom axiom_exp_O : forall n : Nat, equal Nat (exp n O) (S O). Axiom axiom_exp_S : forall n m : Nat, equal Nat (exp n (S m)) (times (exp n m) n).

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Conclusion

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Conclusion

Dedukti D[STT∀βδ] D[CiCω] OpenTheory Matita Coq Lean ? PVS

  • A relatively weak logic: STT∀βδ
  • An automatic translation of a library to 5 other proof systems

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Future work

  • Sharing the aritmetic library to other systems (Agda, Idris,...)
  • Developing an encylopedia of proofs: Logipedia
  • A standardization of an arithmetic library?

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Future work

  • Sharing the aritmetic library to other systems (Agda, Idris,...)
  • Developing an encylopedia of proofs: Logipedia
  • A standardization of an arithmetic library?

Contributions are welcome! https://github.com/Deducteam/Logipedia

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