Generalization of Declared Variables

Variables to be generalized can declared with the keyword variable. For example:

postulate
      Con : Set

variable
      Γ Δ Θ : Con

Declared variables are automatically generalized in type signatures, module telescopes and data and record type parameters and indices. For example,

postulate
      Sub : Con  Con  Set

      id  : Sub Γ Γ
  --  -- equivalent to
  --  id  : {Γ : Con} → Sub Γ Γ

      _∘_ : Sub Θ Δ  Sub Γ Θ  Sub Γ Δ
  --  -- equivalent to
  --  _∘_ : {Γ Δ Θ : Con} → Sub Θ Δ → Sub Γ Θ → Sub Γ Δ

Note that each type signature has a separate copy of its declared variables, so id and _∘_ refer to two different Γ named variables.

When generalizing data type parameters and indicies a variable is turned into an index if it’s only mentioned in indices and into a parameter otherwise. For instance,

variable
  n  : Nat

data Vec (A : Set) : Nat  Set where
  []  : Vec A 0
  _∷_ : A  Vec A n  Vec A (suc n)

variable
  A  : Set
  x  : A
  xs : Vec A n

-- Here `A` will be a parameter and `n` an index. That is,
-- data All {A : Set} (P : A → Set) : {n : Nat} → Vec A n → Set
data All (P : A  Set) : Vec A n  Set where
  []  : All P []
  _∷_ : P x  All P xs  All P (x ∷ xs)

The following rules are used to place the generalized variables:

  • Generalized variables are placed in the front of the type signatures.
  • Variables mentioned eariler are placed before variables mentioned later. (The dependencies between the variables are obeyed. The current implementation uses “smallest-numbered available vertex first” topological sorting to determine the exact order.)

variable allows metavariables in the type of declared variables. For example:

postulate
      Ty  : Con  Set
      _,_ : (Γ : Con)  Ty Γ  Con

variable
      A : Ty _       -- note the underscore here

postulate
      π₁ : Sub Γ (Δ , A)  Sub Γ Δ
  --  -- equivalent to
  --  π₁ : {Γ Δ : Con}{A : Ty Δ} → Sub Γ (Δ , A) → Sub Γ Δ
  --  -- note that the metavariable was solved with Δ

Note that each type signature has a separate copy of such metavariables, so the underscore in Ty _ can be solved differently for each type signature which mentions A.

Unsolved metavariables originated from variable are generalized. For example:

variable
      σ δ ν : Sub _ _   -- metavariables: σ.1, σ.2, δ.1, δ.2, ν.1, ν.2

postulate
      ass : (σ ∘ δ) ∘ ν ≡ σ ∘ (δ ∘ ν)
  --  -- equivalent to
  --  ass : {σ.1 σ.2 δ.1 ν.1 : Con}
  --        {σ : Sub σ.1 σ.2}{δ : Sub δ.1 σ.1}{ν : Sub ν.1 δ.1}
  --      → (σ ∘ δ) ∘ ν ≡ σ ∘ (δ ∘ ν)

Note that δ.2 was solved with σ.1 and ν.2 was solved with δ.1. If two generalizable metavariables can be solved with each-other then the metavariable defined later will be eliminated.

Hierarchical names like δ.2 are used so one can track the origin of the metavariables. Currently it is not allowed to use hierarchical names when giving parameters to functions, see Issue #3280 <https://github.com/agda/agda/issues/3280>.

Name hints of parameters are used for naming generalizable metavariables too:

postulate
      Sub' : (Γ Δ : Con)  Set   -- name hints for parameters of Sub'

variable
      θ : Sub' _ _   -- metavariables: θ.Γ, θ.Δ

If a generalizable metavariable M is solved with term T then M is not generalized, but metavariables in T became candidates for generalization.

It is allowed to nest declared variables. For example:

variable
       : Level     -- let ℓ denote a level
      S : Set-- let A denote a set at a level ℓ

postulate
      f : S  Set--  -- equivalent to
  --  f : {ℓ ℓ' : Level}{S : Set ℓ'} → S → Set ℓ

Issues related to this feature are marked with generalize in the issue tracker: https://github.com/agda/agda/labels/generalize