Coercive subtyping: PAL and beyond

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Coercive subtyping PAL and beyond

• Zhaohui Luo
• Dept of Computer Science
• Royal Holloway, Univ of London
• (In collaboration with Sergei Soloviev)

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I. Dependent type theories

• Type theory
• Curry-Howard principle and canonical objects
• Dependent TTs
• Dependent types
• Inductive types (nats, lists, trees, ordinals,
  )
• (e.g., universes for reflection)
• Existing (intensional) TTs
• Martin-Löfs TT (predicative)
• CID and ECC/UTT (impredicative)

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A simple example

• Type of natural numbers
• N Type
• Introduction rules (canonical objects)
• 0 N
• n1 N n N
• Elimination operator
• for all C Nat?Type, c C(0) and f
  (nNat)C(n)?C(n1),
• rec(C,c,f,n) C(n)
• plus the computation rules
• rec(C,c,f,0) c C(0)
• rec(C,c,f,n1) f(n,rec(C,c,f,n)) C(n1)

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Technology and applications

• Proof technology based on type theories
• Proof assistants ALF/Agda, Coq, Lego, NuPRL,
  Plastic,
• Applications
• Formalisation of mathematics
• Fundamental Theorem of Algebra, Four-colour
  Theorem,
• Program verification
• E.g., security protocols
• Dependently-typed programming
• Cayenne, DML, Epigram

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• Subtyping in dependent TTs?

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II. Typing/subtyping two different views

• View of type assignment
• Type assignment
• Objects exist first types are then assigned to
  objects.
• Overloading ?-terms, which reside in different
  types.
• Subtyping subsumption
• a A A?? B
• ------------------
• a B

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• View of canonical objects
• Canonical objects
• Types consist of canonical objects, which do not
  exist without their types.
• Introduction and elimination rules determine
  canonical objects.
• Unique typing
• Subtyping coercive subtyping subtyping as
  abbreviations
• f A?B a0 A0 A0 ?c A
• ---------------------------------
• f(a0) f(c(a0)) B

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• Equivalence between subsumption and coercion
• Only for simple systems (eg, for simply typed
  lambda calculus Mitchell)
• Difficulty of subsumption with canonical objects
• a A ? B ? a B (Q Is object a
  canonical in B?)
• If so, we would in general require bounded
  quantification
• ?A?B ?CB?Type ?xB instead of ?CB?Type
  ?yB
• c.f., constructor subtyping Coquand, Barthe
  not well-behaved for canonical inhabitants
• Two traditional approaches to subtyping
• Subset/injection (e.g., Even ? N)
• Inheritance/projection (e.g., record subtyping)
• Coercive subtyping
• Solves the above problem with subsumption
• Subsumes the approaches of injection and
  projection

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III. Coercive subtyping

• Framework of coercive subtyping
• Formulated in (meta-level) logical framework LF
• A ltc B Type and K ltc K
• Coherence ? conservativity Soloviev Luo
• Transitivity elimination Y Luo, Z Luo
  Soloviev
• Strong/weak transitivity
• Implementations in proof systems
• Coq Saibi, Lego Bailey, Plastic Callaghan
• Applications
• Proof development (e.g., Bailey)
• Overloading via coercive subtyping
• Dependently-typed programming

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Examples

• Simple coercions
• Even ltc N with c e0 ? 0 and e1(x) ? c(x)2
• where Even has canonical objects e0 and e1(x)
• ?(A,B) lt?1 A with ?1 (a,b) ? a
• Coercions between parameterised inductive types
• List(A) ltmap(c) List(B) for A ltc B
• Parameterised coercions
• Vect(n) ltc(n) List with c(n) lta1,, angt ? a1,
  , an
• Dependent coercions Luo Soloviev
• x List(A) ltc Vect(x) with c a1, , an ?
  lta1,, angt
• x ?(A,B) lt?2 B(?1(x)) with ?2 (a,b) ? b

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III. Coercive subtyping in PAL

• PAL
• Lambda-free logical framework
• Basic concepts in type theory types/objects,
  families of types/objects
• (not families of families of )
• Exactly captured in PAL
• Formally, (?)T where ? ? x1K1, , xnKn and T ?
  Type/El(A)
• rather than arbitrarily nested (x1K1)(x2K2)
  as in other LFs
• Parameterisation and definitions
• as basic notions (instead of ?)
• universal tools in everyday mathematical work
• Adequate for specification of type theories
• E.g., Martin-Löfs type theory, UTT,
• Related work TF a typed framework Aczel,
  Adams

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• Implementations of PAL in UK EPSRC project
  Pythagoras
• Y Luos system
• R Pollacks implementation
• Use of PAL in study of formalisation/foundations
  of math
• Logic-enriched type theories in PAL
• Mathematical pluralism (e.g. Weyls predicative
  math)

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• Coercive subtyping in PAL
• A ?c B Type with c A?B
• (?)T ?c (?)T with c (f(?)T, ?)T
• ? ?? ?
• as abbreviation of a sequence of judgements
• ? being a sequence of coercions
• Extension to dependent coercions is
  straightforward.
• Meta-theory
• Conservativity (detailed checking in progress)
• Remarks
• Implementation
• Using the relationship between coercive subtyping
  and type casting
• Useful in proof development etc.