Implementing a Modal Dependent Type Theory Daniel Gratzer 0 Jonathan Sterling 1 Lars Birkedal 0 August 21, 2019 ICFP ’19 0 Aarhus University 1 Carnegie Mellon University 0
Modalities an idempotent monad for a left adjoint. A Category Theorist 1 We want to add a single modality MLTT, � . Γ ⊢ M : � A � M " : " A and M only mentions variables of the shape � B • In staged programming, � A represents precomputed values. • In modal FRP, � A represents stable types. • In distributed programming, � A represents globally available values. � is just a comonad with
Our Contribution: MLTT We contribute MLTT , a dependent type theory with... decidability of type-checking for it. We have constructed a precise syntactic account of MLTT , and proved the 2 • the box modality, � A • dependent sums, Σ( A , B ) With both β and η • dependent products, Π( A , B ) • natural numbers, nat • intensional identity types, Id ( A , M , N ) • a cumulative hierarchy of universes, U 0 , U 1 ...
Typical Problems with Modalities tm/lock?! 1 Prawitz 1967 3 We could imagine just dropping all local variables when constructing � A : � Γ ⊢ M : A � Γ , ∆ ⊢ box ( M ) : � A In this case box ( M ) cannot commute with substitution: box ( M )[ N / x ] could be well-typed while box ( M [ N / x ]) is ill-typed! We can try versions of this rule, 1 but we’ll opt for another approach.
M Adding Judgmental Structure Crucially, later on we are able to unlock the context: 2 Clouston 2018 A A M tm/unlock 4 tm/lock tm/var Instead of dropping part of the context we can lock it away: (Contexts) We’ll incorporate Fitch-style judgmental structure 2 to handle � A : Γ · | Γ , x : A | Γ . � Γ . ⊢ M : A Γ = Γ 0 , x : A , Γ 1 � Γ 1 Γ ⊢ [ M ] : � A Γ ⊢ x : A
Adding Judgmental Structure tm/lock 2 Clouston 2018 tm/unlock Crucially, later on we are able to unlock the context: tm/var (Contexts) Instead of dropping part of the context we can lock it away: 4 We’ll incorporate Fitch-style judgmental structure 2 to handle � A : Γ · | Γ , x : A | Γ . � Γ . ⊢ M : A Γ = Γ 0 , x : A , Γ 1 � Γ 1 Γ ⊢ [ M ] : � A Γ ⊢ x : A Γ ⊢ M : � A Γ ⊢ [ M ] : A
Adding Judgmental Structure tm/var 2 Clouston 2018 respect substitution! Not obvious, but these rules tm/unlock Crucially, later on we are able to unlock the context: 4 Instead of dropping part of the context we can lock it away: tm/lock (Contexts) We’ll incorporate Fitch-style judgmental structure 2 to handle � A : Γ · | Γ , x : A | Γ . � Γ . ⊢ M : A Γ = Γ 0 , x : A , Γ 1 � Γ 1 Γ ⊢ [ M ] : � A Γ ⊢ x : A Γ ⊢ M : � A Γ ⊢ [ M ] : A
dup A x A Small Programming Break Programs dup A A A Holes 5 How does our intuition for � A square with [ − ] and [ − ] ? extract A : � A → A extract A ( x ) � [ x ]
A Small Programming Break Programs Holes 5 How does our intuition for � A square with [ − ] and [ − ] ? extract A : � A → A extract A ( x ) � [ x ] dup A : � A → �� A x : � A ⊢ ? : �� A dup A ( x ) � ?
A Small Programming Break Programs Holes 5 How does our intuition for � A square with [ − ] and [ − ] ? extract A : � A → A extract A ( x ) � [ x ] dup A : � A → �� A x : � A , ⊢ ? : � A dup A ( x ) � [?]
A Small Programming Break Programs Holes 5 How does our intuition for � A square with [ − ] and [ − ] ? extract A : � A → A extract A ( x ) � [ x ] dup A : � A → �� A x : � A , , ⊢ ? : A dup A ( x ) � [[?] ]
A Small Programming Break Programs Holes 5 How does our intuition for � A square with [ − ] and [ − ] ? extract A : � A → A extract A ( x ) � [ x ] dup A : � A → �� A x : � A ⊢ ? : � A dup A ( x ) � [[[?] ] ]
A Small Programming Break Programs Holes 5 How does our intuition for � A square with [ − ] and [ − ] ? extract A : � A → A extract A ( x ) � [ x ] dup A : � A → �� A dup A ( x ) � [[[ x ] ] ]
M M 3 Clouston 2018 and Birkedal, Clouston, Mannaa, Møgelberg, Pitts, Spitters 2019 A M A A M A Making Hard Choices: Defjnitional Equalities for MLTT The premises of these rules are subtle and important! Notice, no commutating conversions, this is a win from the Fitch style. 3 tm/lock-unlock tm/unlock-lock 6 We are able to equip � A with both a β and η rule in MLTT : Γ . ⊢ M : A Γ ⊢ M : � A Γ ⊢ [[ M ] ] = M : A Γ ⊢ M = [[ M ] ] : � A
Making Hard Choices: Defjnitional Equalities for MLTT tm/unlock-lock tm/lock-unlock Notice, no commutating conversions, this is a win from the Fitch style. 3 The premises of these rules are subtle and important! � 3 Clouston 2018 and Birkedal, Clouston, Mannaa, Møgelberg, Pitts, Spitters 2019 6 We are able to equip � A with both a β and η rule in MLTT : Γ . ⊢ M : A Γ ⊢ M : � A Γ ⊢ [[ M ] ] = M : A Γ ⊢ M = [[ M ] ] : � A Γ . ⊢ M : A = ⇒ Γ ⊢ M : A Γ ⊢ [[ M ] ] : � A = ⇒ Γ ⊢ M : � A
Taking Stock What do we have at this point? Big remaining question: can we implement this? 7 • MLTT : a declarative modal dependent type theory. • We can prove the expected admissibilities: substitution, presupposition, ... • As well as modal admissibilities: lock contraction, strengthening... These are important checks to ensure that MLTT behaves well.
Taking Stock What do we have at this point? Complication: non-local and sensitive to extensions. Big remaining question: can we implement this? 7 • MLTT : a declarative modal dependent type theory. • We can prove the expected admissibilities: substitution, presupposition, ... • As well as modal admissibilities: lock contraction, strengthening... These are important checks to ensure that MLTT behaves well.
Taking Stock What do we have at this point? Big remaining question: can we implement this? 7 • MLTT : a declarative modal dependent type theory. • We can prove the expected admissibilities: substitution, presupposition, ... • As well as modal admissibilities: lock contraction, strengthening... These are important checks to ensure that MLTT behaves well.
Many of these proofs rely on the admissiblities we established! Implementing a Type Theory: A General Recipe 8 The process of implementing some type theory T might follow these steps: 1. Construct a bidirectional syntax for T : T ⇆ . 2. Prove that T admits a normalization theorem. 3. Conclude that T enjoys decidable conversion. 4. Prove that T ⇆ enjoys decidable type-checking. 5. Prove that every term of T is convertible with a term from T ⇆ . 6. Conclude that T ⇆ presents T and is implementable.
Many of these proofs rely on the admissiblities we established! Implementing a Type Theory: A General Recipe 8 The process of implementing some type theory T might follow these steps: 1. Construct a bidirectional syntax for T : T ⇆ . 2. Prove that T admits a normalization theorem. 3. Conclude that T enjoys decidable conversion. 4. Prove that T ⇆ enjoys decidable type-checking. 5. Prove that every term of T is convertible with a term from T ⇆ . 6. Conclude that T ⇆ presents T and is implementable.
Many of these proofs rely on the admissiblities we established! Implementing a Type Theory: A General Recipe 8 The process of implementing some type theory T might follow these steps: 1. Construct a bidirectional syntax for T : T ⇆ . 2. Prove that T admits a normalization theorem. 3. Conclude that T enjoys decidable conversion. 4. Prove that T ⇆ enjoys decidable type-checking. 5. Prove that every term of T is convertible with a term from T ⇆ . 6. Conclude that T ⇆ presents T and is implementable.
Many of these proofs rely on the admissiblities we established! Implementing a Type Theory: A General Recipe 8 The process of implementing some type theory T might follow these steps: 1. Construct a bidirectional syntax for T : T ⇆ . 2. Prove that T admits a normalization theorem. 3. Conclude that T enjoys decidable conversion. 4. Prove that T ⇆ enjoys decidable type-checking. 5. Prove that every term of T is convertible with a term from T ⇆ . 6. Conclude that T ⇆ presents T and is implementable.
Many of these proofs rely on the admissiblities we established! Implementing a Type Theory: A General Recipe 8 The process of implementing some type theory T might follow these steps: 1. Construct a bidirectional syntax for T : T ⇆ . 2. Prove that T admits a normalization theorem. 3. Conclude that T enjoys decidable conversion. 4. Prove that T ⇆ enjoys decidable type-checking. 5. Prove that every term of T is convertible with a term from T ⇆ . 6. Conclude that T ⇆ presents T and is implementable.
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