A Logical Framework with Dependently Typed Records Thierry Coquand, - PDF document

A Logical Framework with Dependently Typed Records Thierry Coquand, Randy Pollack, Makoto Takeyama April 16, 2003 (809) A Logical Framework with Dependently Typed Records Thierry Coquand, Slide 1 Robert Pollack, Makoto Takeyama April 16, 2003 1

2 Long Term Goal: Precise explanation of mathematical vernacular. � Experiment with dependently typed records as a notation for mathematical structure. { Manifest fields { Structural subtyping with ’ notation to add information to signatures { ‘ Slide 2 { Signature strengthening ringSig = h G : grpSig ; M : monSig with rr = G : rr ; : : : i : � Typed equality. { � and surjective pairing { respects subtyping � Direct and simple implementation. This talk Restrict to a simple core similar to Martin-L¨ of’s framework. � PER semantics { Categorical judgements of Type Theory s.t. (up to � -expansion) Slide 3 � Well-typed objects are normalising � Equality is decidable { Extend to hypothetical judgements of Type Theory � Implementable rules for a logical framework. � Extend core system to subtyping . This core framework can be extended with definitions, structures and signatures, manifest fields, . . . See our paper.

3 Syntactic Objects and Syntactic Types � x , y , range over identifiers , I . � objects , O , are untyped � -terms. M ; N ::= x j M M j �x:M { Equality on objects is � -conversion ( ' ). Slide 4 { An object is neutral iff it is normalisable and of the form � ::= x j � M : � The category of syntactic types A; B ::= El M j fun A x:B j ? { Objects in ? are “names” of types; { for : ? , M is the type named by M . M El Categorical Judgements: Outline � Simultaneously define 1. A PER of intensional equality on the set of syntactic types. Write = B . A Write A 2 T yp e for A = A . Slide 5 2. For A 2 T yp e , a PER, A , on objects. Write M = N : A . Write : A for = : A . M M M � Define eta-expansion � Key normalization theorem.

4 Categorical Judgements: the definition � = ? . ? = : ? iff M and N are neutral and � -convertible. M N � = N whenever = : ? . El M El M N = : M iff 1 and 2 are neutral and N N El N N 1 2 � -convertible. Slide 6 � = 2 whenever fun A x :B fun A x :B 1 1 1 2 2 { A = A 2 , 1 { = : = ) [ M ℄ = [ M ℄ . M M A B B 1 2 1 1 1 2 2 M = M : fun A x:B iff 1 2 N = N : A = ) M N = M N : B [ M ℄ . 1 2 1 1 2 2 1 If A = B then A and B are extensionally equal. The converse is not true. Eta-Expansion Define an operation of � -expansion at type A (written f A g ). � : f A g M is normalising. M A will imply � Slide 7 � f ? g = �x:x � f El M g = �x:x � f fun A x:B g = �u:�z :� f B [ z b ℄ g ( u z b ) where z b = � f A g z where u and z are distinct and not free in � f A g or B .

5 Key Theorem: Normalization Theorem Let A 2 T yp e . 1. � f A g � : A , where � is neutral. : f A g M is normalisable. 2. If M A then � 3. If : A then = f A g M : A . M M � Slide 8 4. If = : A then f A g M ' f A g M 2 . M M � � 1 2 1 The four parts are proved simultaneously by induction on the proof that 2 T yp e . A Corollary Let A 2 T yp e , M : A and M : A . 1 2 f A g M ' f A g M = : 1. If � � 2 then M M A . 1 1 2 2. The relation = : A is decidable. M M 1 2 Hypothetical Judgements: Outline � Introduce environments , � , contexts , C . � Define a judgement � = � : C . 1 2 � Simultaneously define hypothetical judgements { C valid , { A = A [ C ℄ , 1 2 Slide 9 { = : [ C ℄ . M M A 1 2 For C valid , � = � : C , A = A [ C ℄ and 1 2 1 2 M = M : A [ C ℄ are PERs. 1 2 � Show that hypothetical judgements satisfy the rules of Type Theory. � Give conditions for M = M : A [ C ℄ to be decidable. 1 2 � Define a relation ` = 2 of syntactic type equality that is C A A 1 decidable and sound for the sematintic relation A = A [ C ℄ . 1 2

6 Contexts and Environments Contexts C ::= j C; x : A ( O is the empty context.) O Write x 2 C if x : A in C for some A . In writing C; x : A we assume x 62 C . Environments Slide 10 � An environment, � , is a function I ! O . � � 0 is the identity environment. � Environments are applied as simultaneous substitutions: M � , A� . � Write ( �; x = M ) for the update of � , defined by ( �; x = M )( x ) = M ; ( �; x = M )( y ) = � ( y ) if y 6 = x: Equal Environments Inductively define a judgement of form = : C : � � 1 2 Slide 11 � = � : O 1 2 � = � : C A� 2 T yp e � x = � x : A� 1 2 1 1 2 1 � = � : C; x : A 1 2 Write � : C for � = � : C .

7 Hypothetical Judgements Defined Simultaneously define three judgement forms: validity x 62 C A = A [ C ℄ valid C; x : A valid O type equality (write A t yp e [ C ℄ for A = A [ C ℄ ) Slide 12 C valid 8 � ; � : � = � : C = ) A � = A � 1 2 1 2 1 1 2 2 A = A [ C ℄ 1 2 object equality in a type (write : [ C ℄ for = : [ C ℄ ) M A M M A A t yp e [ C ℄ 8 � ; � : � = � : C = ) M � = M � : A� 1 2 1 2 1 1 2 2 1 M = M : A [ C ℄ 1 2 Properties of Hypothetical Judgements type formation and type equality C valid M = N : ? [ C ℄ A = A [ C ℄ B = B [ C; x : A ℄ 1 2 1 2 1 ? t yp e [ C ℄ El M = El N [ C ℄ fun A x:B = fun A x:B [ C ℄ 1 1 2 2 objects C; x : A valid M : B [ C; x : A ℄ x : A [ C; x : A ℄ �x:M : fun A x:B [ C ℄ Slide 13 : [ C ℄ : [ C ℄ M fun A x:B N A M N : B [ N ℄ [ C ℄ type conversion M = N : A [ C ℄ A = B [ C ℄ M = N : B [ C ℄ weakening = [ C ℄ x : = : [ C ℄ x : B B C; A valid M N B C; A valid 1 2 = [ C; x : A ℄ = : [ C; x : A ℄ B B M N B 1 2

8 Deciding Hypothetical Equality � We saw that if M ; M : A then M = M : A is decidable. 1 2 1 2 � Lift this property to hypothetical judgements: { need the “most general environment”, { intuitively, the identity environment, 0 . � � By the Key Theorem, we need to � -expand 0 at C : � � f O g � = � Slide 14 � f C; x : A g � = ( � 0 ; x = � f A� 0 g ( �x )) where � 0 = � f C g � Lemma Write � C for � f C g � 0 . � If valid then : C . � Æ = f C g � . C � � � � C C � If : [ C ℄ and : [ C ℄ then M A M A 1 2 M = M : A [ C ℄ iff � f A� g M � ' � f A� g M � 1 2 C 1 C C 2 C which is decidable. Approximating Type Equality Define a syntactic relation of shape ` = 2 which is C A A 1 = [ C ℄ . decidable and sound for the semantic relation A A 1 2 M = M : ? [ C ℄ 1 2 C ` ? = ? C ` El M = El M 1 2 Slide 15 ` = x : ` = C A A C; A B B 1 2 1 1 2 ` = C fun A x:B fun A x:B 1 1 2 2 Lemma 1. If [ C ℄ and [ C ℄ then ` = 2 is A t yp e A t yp e C A A 1 2 1 decidable. 2. If valid and ` = 2 then = [ C ℄ . C C A A A A 1 1 2

9 A Logical Framework in Syntax: Outline We give a concrete implementation. � Syntax of expressions and expression contexts Slide 16 � Judgement forms � Typechecking rules { syntax directed { terminating { sound Expressions The syntax of expressions and expression contexts: e ::= z j e e j [ z : e ℄ e j * j El e j f z : e g e j e - > e � ::= j � , x : e ( H is the empty context) H Judgement Forms Slide 17 Two judgement forms are defined simultaneously, � C ` e ) A , (expression e is interpreted in C as type A ) � C ` e ) M : A , (expression e is interpreted in C as object M having type A ) while a third can be defined afterwards. � � ) C , ( � is interpreted as the valid context C )

10 Typechecking type formation C ` e ) M : ? C ` e ) A C ` e ) B 1 2 C ` * ) ? C ` El e ) El M C ` e - > e ) A ! B 1 2 C ` e ) A C; x : A ` e ) B 1 2 C ` f x : e g e ) fun A x:B 1 2 Slide 18 objects x : A in C C ` e ) A C; x : A ` e ) M : B 1 2 C ` x ) x : A C ` [ x : e ℄ e ) �x:M : fun A x:B 1 2 C ` e ) M : fun A x:B C ` e ) M : A C ` A = A 1 1 1 2 2 2 1 2 C ` e e ) M M : B [ M ℄ 1 2 1 2 2 validity � ) C C ` e ) A ) � ; x : e ) C; x : A H O Correctness and Termination of Typechecking � If ` ) A and valid then [ C ℄ . C a C A t yp e � If C ` e ) M : A and C valid then M : A [ C ℄ . � If � ) C then C valid . Slide 19 � Typechecking is decidable. “Eta-conversion” of expressions v:{A:*}El A . � Suppose v and [A:*℄(v A) are indistinguishable by � The expressions typechecking.