. . . . . . . . . . . . . . . . . . . . TYPES 2014, May 12 TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . 1 / 24 Objects and subtyping in the λ Π -calculus modulo Ali Assaf, Raphaël Cauderlier , Catherine Dubois Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . Motivations Functional Pure Type Systems Proof assistants: Coq, HOL, FoCaLize Theorem provers: Zenon, iProver Object calculi have type systems with (object) subtyping Subtyping is a common feature of type systems, also present in Coq (universes) TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . 2 / 24 . . . . . The λ Π -calculus modulo has been designed to encode other calculi We use λ Π -calculus modulo rewriting to study OOL semantics How can we translate object mechanisms in the λ Π -calculus modulo? The λ Π -calculus modulo lacks subtyping Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . Related work . Several deep encodings: Cardelli (1984), Pierce, Turner and Hofmann (1993-1995), Bruce (1993), Abadi, Cardelli and Viswanathan (1996) Implemented in Yarrow (1997): a proof assistant with object subtyping Objects , Springer Verlag, 1996 Deep encodings in Coq, focus on proving properties on the type system by Gillard and Despeyroux (1999): reasoning on binders encoded via DeBrujn indices and Liquori (2007): proof of the subject-reduction theorem In Isabelle/HOL: deep formalisation of class-based languages (parts of Java and Scala) with extensible records: Klein and Nipkow (2005), Foster and Vytiniotis (2006) TYPES 2014, May 12 . . . . . . . . . . . . . . . . 3 / 24 . . . . . . . . . . . . . . In System F ω <: (polymorphism, type operators and subtyping) Object calculi (a.k.a ς -calculi) from Abadi and Cardelli, A Theory of Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . This work Shallow embedding semantically equal terms, types or proofs should not be distinguishable after the encoding expected efficiency readability TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 24 Encoding of an object calculus: the simply-typed ς -calculus In the λ Π -calculus modulo Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . . Outline 1 2 3 TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 24 The λ Π -calculus modulo and Dedukti The simply-typed ς -calculus Explicit subtyping in the λ Π -calculus modulo Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . . . . (Conv) TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . 6 / 24 . . . . . The λ Π -calculus modulo The λ Π -calculus is a typed λ calculus with dependent types The λ Π -calculus modulo, introduced by Cousineau and Dowek in 2007, extends the λ Π -calculus with a rewrite system R . Γ ⊢ t : A A ≡ β R B Γ ⊢ t : B Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . Dedukti It is a free software, available at https://www.rocq.inria.fr/deducteam/Dedukti/ Dependent types Rewriting on terms and types Partial functions and proofs Non-linear pattern-matching TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 24 Type-checker for the λ Π -calculus modulo Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . . of Objects , 1996 Functional semantics (imperative semantics also studied) Model of both class-based and object-based languages No termination guaranted by typing Structural subtyping TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 24 The simply-typed ς -calculus: Abadi and Cardelli, A Theory Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . Syntax and semantics Types labels are unordered Terms t, u ::= t.l Operational semantics t.l j TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . 9 / 24 . . . . . A ::= [ l i : A i ] i=1..n [ l i = ς (x : A) t i ] i=1..n t.l ⇐ ς (x : A) u (t.l ⇐ u) abbreviates (t.l ⇐ ς (x : A) u) where x ̸∈ FV(u). (l = u) abbreviates (l = ς (x : A) u) where x ̸∈ FV(u). A := [ l i : A i ] i=1..n t := [ l i = ς (x : A) t i ] i=1..n ↣ t j [t/x] t.l j ⇐ ς (x : A) u [ l j = ς (x : A) u, l i = ς (x : A) t i ] i =1..n, i ̸ = j ↣ Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . Typing and subtyping (obj) (select) (update) A <: B (subsume) TYPES 2014, May 12 . . . . . . . . . . . . . . . 10 / 24 . . . . . . . . . . . . A := [ l i : A i ] i=1..n ∀ i=1..n Γ , x : A ⊢ t i : A i Γ ⊢ t : A Γ ⊢ [ l i = ς (x : A) t i ] i=1..n : A Γ ⊢ t.l i : A i Γ ⊢ t : A Γ , x : A ⊢ u : A i Γ ⊢ t.l i ⇐ ς (x : A) u : A [ l i : A i ] i=1..n+m <: [ l i : A i ] i=1..n Γ ⊢ t : A Γ ⊢ t : B Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . . . Example: Encoding of booleans "then" and "else" methods are updated before "if" is selected TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . 11 / 24 . . . . . Bool A := [if : A, then : A, else : A] true A := [if = ς (self : A) self.then, then = ς (self : A) self.then, else = ς (self : A) self.else] false A := [if = ς (self : A) self.else, then = ς (self : A) self.then, else = ς (self : A) self.else] if A b then t else e := ((b.then ⇐ t).else ⇐ e).if Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . Subtyping example RomCell := [ get : nat ] PromCell := PromCell <: RomCell myCell : PromCell := [ get = 0, TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 / 24 [ get : nat, set : nat → RomCell ] set = ς (self : PromCell) λ (n : nat) self.get ⇐ n ] myCell.set(42).get ↣ ∗ 42 Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . . . Types and objects are translated as association lists The operational semantics is translated to rewrite rules Subtyping is explicit TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . 13 / 24 Translation scheme from simply-typed ς -calculus to λ Π -calculus modulo Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . . Explicit subtyping Convertibility is a symmetric relation We cannot rewrite A to B whenever A <: B because that would make both types equal Hence we ask the user to provide explicit coercions (subtyping annotations) TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 14 / 24 In the λ Π -calculus modulo, each term has at most one type modulo the rewrite system + β conversion Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . . . Translation of types Types are translated by normalized association lists Equality and subtyping relations on types are decidable: A <: (l, B 1 ) :: B 2 TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . 15 / 24 . . . . . A = A ֒ → true []= ( _ , _ ) :: _ ֒ → false A <: [] ֒ → true ( _ , _ ) :: _ = [] ֒ → false ֒ → B 1 = assoc A l ∧ A <: B 2 (l 1 , A 1 ) :: B 1 = (l 2 , A 2 ) :: B 2 ֒ → l 1 = l 2 ∧ A 1 = A 2 ∧ B 1 = B 2 Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
. . . . . . . . . . . . . . . . Translation of objects Objects are also translated by association lists with labels in the same order than in the corresponding type an object of type A is something of the form sublists are not well-typed objects to construct objects, we need to consider (ill-typed) objects defined on subsets of dom(A) to coerce objects, we need to consider (ill-typed) objects with methods typed by (assoc B). TYPES 2014, May 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 / 24 [l = ς (x : A) (t : assoc A l)] l ∈ dom(A) ⇒ A pre-object of type (A, f, D) is something of the form [l = ς (x : A) (t : f l)] l ∈ D Ali Assaf, Raphaël Cauderlier , Catherine Dubois Objects and subtyping in the λ Π -calculus modulo
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