Formalisation in Constructive Type Theory of Barendregt’s Variable Convention for Generic Structures with Binders Ernesto Copello 1 Nora Szasz 2 Álvaro Tasistro 2 1 Department of Computer Science The University of Iowa, USA 2 Facultad de Ingeniería Universidad ORT Uruguay June 22, 2018 1
Outline • We introduce a universe of regular datatypes with variable binding information with: – a first-order named syntax interpretion – usual formation and elimination operators – operations and predicates specific to variables (swapping, free variables, fresh binders, etc) – an α -equivalence relation based on name-swapping. – iteration and induction principles which capture the Barendregt’s Variable Convention • We instantiate λ -Calculus and System F, deriving: – almost free substitution operations and α -conversion lemmas – substitution composition lemma • The whole work is carried out in Constructive Type Theory and machine-checked by the system Agda. 1
Regular Trees Types with Binders • Functor datatype: introduces the codes of functors. • � _ � function: gives the interpretation of codes. • µ datatype: represents the fixpoint of some given F functor. mutual data Functor : Set 1 where � _ � : Functor → Set → Set � |1| � _ = ⊤ |1| : Functor � |R| � A = A |R| : Functor � |E| B � _ = B |E| : Set → Functor � |Ef| F � _ = µ F |Ef| : Functor → Functor � F |+| G � A = � F � A ⊎ � G � A _|+|_ : Functor → Functor → Functor � F |x| G � A = � F � A × � G � A _|x|_ : Functor → Functor → Functor � |v| S � _ = V |v| : Sort → Functor � |B| S G � A = V |B| : Sort → Functor → Functor × � G � A data µ ( F : Functor) : Set where � _ � : � F � ( µ F ) → µ F 2
Regular Trees Types with Binders • Functor datatype: introduces the codes of functors. • � _ � function: gives the interpretation of codes. • µ datatype: represents the fixpoint of some given F functor. mutual data Functor : Set 1 where � _ � : Functor → Set → Set � |1| � _ = ⊤ |1| : Functor � |R| � A = A |R| : Functor � |E| B � _ = B |E| : Set → Functor � |Ef| F � _ = µ F |Ef| : Functor → Functor � F |+| G � A = � F � A ⊎ � G � A _|+|_ : Functor → Functor → Functor � F |x| G � A = � F � A × � G � A _|x|_ : Functor → Functor → Functor � |v| S � _ = V |v| : Sort → Functor � |B| S G � A = V |B| : Sort → Functor → Functor × � G � A data µ ( F : Functor) : Set where � _ � : � F � ( µ F ) → µ F 2
Regular Trees Types with Binders • Functor datatype: introduces the codes of functors. • � _ � function: gives the interpretation of codes. • µ datatype: represents the fixpoint of some given F functor. mutual data Functor : Set 1 where � _ � : Functor → Set → Set � |1| � _ = ⊤ |1| : Functor � |R| � A = A |R| : Functor � |E| B � _ = B |E| : Set → Functor � |Ef| F � _ = µ F |Ef| : Functor → Functor � F |+| G � A = � F � A ⊎ � G � A _|+|_ : Functor → Functor → Functor � F |x| G � A = � F � A × � G � A _|x|_ : Functor → Functor → Functor � |v| S � _ = V |v| : Sort → Functor � |B| S G � A = V |B| : Sort → Functor → Functor × � G � A data µ ( F : Functor) : Set where � _ � : � F � ( µ F ) → µ F 2
Regular Trees Types with Binders • Functor datatype: introduces the codes of functors. • � _ � function: gives the interpretation of codes. • µ datatype: represents the fixpoint of some given F functor. mutual data Functor : Set 1 where � _ � : Functor → Set → Set � |1| � _ = ⊤ |1| : Functor � |R| � A = A |R| : Functor � |E| B � _ = B |E| : Set → Functor � |Ef| F � _ = µ F |Ef| : Functor → Functor � F |+| G � A = � F � A ⊎ � G � A _|+|_ : Functor → Functor → Functor � F |x| G � A = � F � A × � G � A _|x|_ : Functor → Functor → Functor � |v| S � _ = V |v| : Sort → Functor � |B| S G � A = V |B| : Sort → Functor → Functor × � G � A data µ ( F : Functor) : Set where � _ � : � F � ( µ F ) → µ F 2
Lambda Calculus Example λ F : Functor � M,N :- λ F = |v| Sort λ TermVars � x |+| |R| |x| |R| � | M N |+| |B| Sort λ TermVars |R| � | λ x . M λ Term : Set λ Term = µ λ F v : V → λ Term v = � _ � ◦ inj 1 _ · _ : λ Term → λ Term → λ Term M · N = � inj 2 (inj 1 ( M , N )) � Ż : V → λ Term → λ Term Ż n M = � inj 2 (inj 2 ( n , M )) � 3
System F Example tyF : Functor � t,r :- tyF = |v| SortFTypeVars � α |+| |R| |x| |R| � | t → r |+| |B| SortFTypeVars |R| � | ∀ α . t tF : Functor � M,N :- tF = |v| SortFTermVars � x |+| |R| |x| |R| � | M N |+| |Ef| tyF |x| |B| SortFTermVars |R| � | λ x : t . M |+| |R| |x| |Ef| tyF � | M t |+| |B| SortFTypeVars |R| � | Λ α . M FType : Set FType = µ tyF FTerm : Set FTerm = µ tF 4
System F Example tyF : Functor � t,r :- tyF = |v| SortFTypeVars � α |+| |R| |x| |R| � | t → r |+| |B| SortFTypeVars |R| � | ∀ α . t tF : Functor � M,N :- tF = |v| SortFTermVars � x |+| |R| |x| |R| � | M N |+| |Ef| tyF |x| |B| SortFTermVars |R| � | λ x : t . M |+| |R| |x| |Ef| tyF � | M t |+| |B| SortFTypeVars |R| � | Λ α . M FType : Set FType = µ tyF FTerm : Set FTerm = µ tF 4
System F Example tyF : Functor � t,r :- tyF = |v| SortFTypeVars � α |+| |R| |x| |R| � | t → r |+| |B| SortFTypeVars |R| � | ∀ α . t tF : Functor � M,N :- tF = |v| SortFTermVars � x |+| |R| |x| |R| � | M N |+| |Ef| tyF |x| |B| SortFTermVars |R| � | λ x : t . M |+| |R| |x| |Ef| tyF � | M t |+| |B| SortFTypeVars |R| � | Λ α . M FType : Set FType = µ tyF FTerm : Set FTerm = µ tF 4
Lambda Calculus Fold Instantiation Example λ F : Functor varsaux : � λ F � N → N � M,N :- λ F = |v| Sort λ TermVars varsaux (inj 1 _) = 1 � x |+| |R| |x| |R| varsaux (inj 2 (inj 1 ( m , n ))) = m + n � | M N |+| |B| Sort λ TermVars |R| � | λ x . varsaux (inj 2 (inj 2 (_ , m ))) = m M vars : µ λ F → N vars = fold λ F varsaux vars function could also be defined generically (for any functor). 5
Fold with Context( µ C ) and a Functorial Return Type( µ H ) Fold instance • adds a c extra argument of type µ C , used by the folded function f as an explicit invariant context through the entire fold operation • the µ H type of the result is an instance of our universe (instead of an arbitrary set as in fold ). foldCtx : { C H : Functor}( F : Functor) → ( µ C → � F � ( µ H ) → µ H ) → µ C → µ F → µ H foldCtx F f c = fold F ( f c ) 6
Lambda Calculus Example: Naive Substitution We derive the naive substitution for the λ -calculus from previous fold instance. Using the cF functor descriptor for the context argument, representing the pair formed by the variable to be replaced and the substituted term. cF = |v| Sort λ TermVars |x| |Ef| λ F substaux : µ cF → � λ F � ( µ λ F) → µ λ F substaux _ (inj 2 (inj 1 ( t 1 , t 2 ))) = t 1 · t 2 substaux _ (inj 2 (inj 2 ( y , t ))) = Ż y t ? substaux � x , N � (inj 1 y ) with x =v y ... | yes _ = N ... | no _ = v y _[_ ≔ _] n : λ Term → V → λ Term → λ Term M [ x ≔ N ] n = foldCtx λ F substaux � x , N � M 7
Primitive Induction fih function receives a predicate P : µ F → Set ,and returns a predicate � G � ( µ F ) → Set , representing P holding in all µ F recursive positions in an element of type � G � ( µ F ). fih : { F : Functor}( G : Functor)( P : µ F → Set) → � G � ( µ F ) → Set fih |1| P tt = ⊤ fih |R| = P e P e fih (|E| B ) = ⊤ P e fih (|Ef| G ) = ⊤ P e fih ( G 1 |+| G 2 ) P (inj 1 e ) = fih G 1 P e fih ( G 1 |+| G 2 ) P (inj 2 e ) = fih G 2 P e fih ( G 1 |x| G 2 ) P ( e 1 , e 2 ) = fih G 1 P e 1 × fih G 2 P e 2 fih (|v| S ) = ⊤ P x fih (|B| S G ) P ( x , e ) = fih G P e 8
Primitive Induction foldmapFh : { F : Functor}( G : Functor)( P : µ F → Set) → (( e : � F � ( µ F )) → fih F P e → P � e � ) → ( x : � G � ( µ F )) → fih G P x foldmapFh |1| P hi tt = tt foldmapFh { F } |R| = hi e (foldmapFh { F } F P hi e ) P hi � e � foldmapFh (|E| B ) = tt P hi b foldmapFh (|Ef| F ) = tt P hi b foldmapFh ( G 1 |+| G 2 ) P hi (inj 1 e ) = foldmapFh G 1 P hi e foldmapFh ( G 1 |+| G 2 ) P hi (inj 2 e ) = foldmapFh G 2 P hi e foldmapFh ( G 1 |x| G 2 ) P hi ( e 1 , e 2 ) = foldmapFh G 1 P hi e 1 , foldmapFh G 2 P foldmapFh (|v| S ) = tt P hi n foldmapFh (|B| S G ) P hi ( x , e ) = foldmapFh G P hi e foldInd : ( F : Functor)( P : µ F → Set) → (( e : � F � ( µ F )) → fih F P e → P � e � ) → ( e : µ F ) → P e foldInd F P hi e = foldmapFh { F } |R| P hi e 9
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