Importing HOL-Light into Coq

Transcription

1 Outlines Importing HOL-Light into Coq Deep and shallow embeddings of the higher order logic into Coq Work in progress Chantal Keller Bejamin Werner 2009 Types meeting Importing HOL-Light into Coq 1 / 24

2 Outlines Introduction What: long term: importing HOL-Light theorems and proofs into Coq short term: encoding the Higher Order Logic into Coq dening and exporting HOL-Light proof terms Why: theoretical interest analysis libraries verication of HOL-Light into Coq Importing HOL-Light into Coq 2 / 24

3 Outlines Double embedding Deep embedding (data-type to represent types and terms): reasoning by induction over the structure Shallow embedding (using Coq types and terms): using the Coq features Importing HOL-Light into Coq 3 / 24

4 Outlines Double embedding Deep embedding (data-type to represent types and terms): reasoning by induction over the structure Shallow embedding (using Coq types and terms): using the Coq features obtaining Coq propositions Importing HOL-Light into Coq 3 / 24

5 Outlines Double embedding Deep embedding (data-type to represent types and terms): reasoning by induction over the structure simple compact Shallow embedding (using Coq types and terms): using the Coq features obtaining Coq propositions Importing HOL-Light into Coq 3 / 24

6 Outlines Double embedding Deep embedding (data-type to represent types and terms): reasoning by induction over the structure simple compact Shallow embedding (using Coq types and terms): using the Coq features obtaining Coq propositions translation function from deep to shallow Importing HOL-Light into Coq 3 / 24

7 Outlines Idea HOL Light + proof recording Coq + classical axioms HOL Light theorem export encoded proof term translation function Coq proposition P proof of P Importing HOL-Light into Coq 4 / 24

8 Outlines Outlines 1 A short presentation of HOL-Light 2 Embedding the higher order logic into Coq 3 Recording and exporting HOL-Light proof terms 4 Conclusion and perspectives Importing HOL-Light into Coq 5 / 24

9 A short presentation of HOL-Light Part I A short presentation of HOL-Light Importing HOL-Light into Coq 6 / 24

10 A short presentation of HOL-Light HOL-Light HOL-Light: proof assistant written by John Harrison et al. in an OCaml top-level higher order classical logic automated tools and pre-proved theorems programmable without compromising soundness simpler logical kernel than HOL Importing HOL-Light into Coq 7 / 24

11 A short presentation of HOL-Light Types and terms Logical framework: simply-typed λ-calculus terms and type variables and constants polymorphism: type schemes all the types must be inhabited theorem: term of type bool under the hypotheses of other terms of type bool Example: no proof terms!x:a.?y:a. x = y Importing HOL-Light into Coq 8 / 24

12 A short presentation of HOL-Light Remarks Example of an inference rule: Γ p q p Γ q where is = bool Constants: main type constants: bool and > main term constants: = : A > A > bool and ε : (A > bool) > A (choice operator) possibility to dene new constants Importing HOL-Light into Coq 9 / 24

13 Embedding the higher order logic into Coq Part II Embedding the higher order logic into Coq Importing HOL-Light into Coq 10 / 24

14 Embedding the higher order logic into Coq Presentation What we want: deep and shallow embeddings translation function from deep to shallow HOL-Light inference rules proof of correctness of these inference rules with respect to semantics Carrying out: inductive Coq data-types type and term a translation function sem_term that maps any term of type Bool onto a term of type Prop (in particular) inductive data-type deriv : set term > term > Prop a proof of: forall G p, deriv G p > has_sem G > has_sem p Importing HOL-Light into Coq 11 / 24

15 Embedding the higher order logic into Coq Types Inductive data-type type: Bool type Num type X idt TVar X type C deft T 1,..., T n type TDef C [T 1 ;... ; T n ] type A, B type A B type idt, deft two sets. Importing HOL-Light into Coq 12 / 24

16 Embedding the higher order logic into Coq Constants Inductive data-type cst: Hand cst Hor cst Himp cst Hnot cst A type A type Htrue cst Hfalse cst Heq A cst Heps A cst A type Hforall A cst A type Hexists A cst Importing HOL-Light into Coq 13 / 24

17 Embedding the higher order logic into Coq Terms Inductive data-type term: c cst Cst c term n N Dbr n term x idv A type Var x A term c defv C type Def c C term u, v term App u v term A type u term Abs A u term idv, defv two sets Importing HOL-Light into Coq 14 / 24

18 Embedding the higher order logic into Coq Translation General idea: types: interface between syntax and semantics translation of a type: T A B A B translation of a term (using dependent types): t, T, t : T T Importing HOL-Light into Coq 15 / 24

19 Embedding the higher order logic into Coq Translation General idea: types: interface between syntax and semantics translation of a type: T? A B? A? B? translation of a term (using dependent types): t, T, t : T T? Importing HOL-Light into Coq 15 / 24

20 Embedding the higher order logic into Coq Translation General idea: types: interface between syntax and semantics translation of a type: T? A B? A? B? translation of a term (using dependent types): t, T, t : T T? a De Bruijn context interpretation functions for variables and denitions Importing HOL-Light into Coq 15 / 24

21 Embedding the higher order logic into Coq Translation General idea: types: interface between syntax and semantics translation of a type: T? A B? A? B? translation of a term (using dependent types): t, T, t : T T? a De Bruijn context interpretation functions for variables and denitions Code Importing HOL-Light into Coq 15 / 24

22 Embedding the higher order logic into Coq Inference rules General idea: inductive data-type deriv : set term > term > Prop a proof of: forall G p, deriv G p > has_sem G > has_sem p has_sem p: p is locally closed p:bool the translation of p is a correct proposition Importing HOL-Light into Coq 16 / 24

23 Embedding the higher order logic into Coq Inference rules General idea: inductive data-type deriv : set term > term > Prop a proof of: forall G p, deriv G p > has_sem G > has_sem p has_sem p: p is locally closed p:bool the translation of p is a correct proposition Code Importing HOL-Light into Coq 16 / 24

24 Embedding the higher order logic into Coq Example!x:A.?y:A. x = y x = A x y : A. x = A y REFL x x : A. y : A. x = A y EXISTS y : A. x =A y x GEN x Importing HOL-Light into Coq 17 / 24

25 Embedding the higher order logic into Coq Example!x:A.?y:A. x = y x = A x y : A. x = A y REFL x x : A. y : A. x = A y EXISTS y : A. x =A y x GEN x Code Importing HOL-Light into Coq 17 / 24

26 Recording and exporting HOL-Light proof terms Part III Recording and exporting HOL-Light proof terms Importing HOL-Light into Coq 18 / 24

27 Recording and exporting HOL-Light proof terms Proof-recording system by S. Obua Challenge: compact proofs short recording time Solution: granularity Statistics: recording the basic HOL-Light proofs (1694 theorems): 3 min Importing HOL-Light into Coq 19 / 24

28 Recording and exporting HOL-Light proof terms Exporting Challenge: small les small number of les Solution: sharing (proofs, types and terms... ) Statistics: exporting the basic HOL-Light proofs (1694 theorems): 14 min '.v' les 2.2 Gb Importing HOL-Light into Coq 20 / 24

29 Conclusion and perspectives Part IV Conclusion and perspectives Importing HOL-Light into Coq 21 / 24

30 Conclusion and perspectives Conclusion HOL-Light: recording proof terms export proofs Coq: Coq representation of HOL-Light data-types standard lemmas (substitution... ) translation function Coq representation of HOL-Light inference rules proof of correctness Importing HOL-Light into Coq 22 / 24

31 Conclusion and perspectives Perspectives Perspectives: nish the interface and the proofs deal with inhabited types, denitions, axioms more ecient Coq data-types more ecient exportation and smaller proof terms user interface scaling up Importing HOL-Light into Coq 23 / 24

32 Thank you for your attention! Any questions? Importing HOL-Light into Coq 24 / 24