{-# OPTIONS --without-K --safe #-} -- Definition of the PER model -- -- The PER model relates two domain values such that the syntactic terms they -- represent are equivalent. Since we are handling MLTT with full a ω universe, we must -- use a feature, induction-recursion, to strengthen the logical power of the -- meta-language. module MLTT.Semantics.PER where open import Data.Nat.Properties open import Lib hiding (lookup) open import MLTT.Semantics.Domain open import MLTT.Semantics.Evaluation open import MLTT.Semantics.Readback open import Relation.Binary using (Rel) Ty : Set₁ Ty = Rel D _ Ev : Set₁ Ev = Rel Env _ -- Two neutral domain values are related if they are read back equal Bot : Dn → Dn → Set Bot c c′ = ∀ n → ∃ λ u → Re n - c ↘ u × Re n - c′ ↘ u -- Two normal domain values are related if they are read back equal Top : Df → Df → Set Top d d′ = ∀ n → ∃ λ w → Rf n - d ↘ w × Rf n - d′ ↘ w -- Two domain values intended to represent types are related if they are read back equal TopT : D → D → Set TopT A B = ∀ n → ∃ λ W → Rty n - A ↘ W × Rty n - B ↘ W -- A PER to model natural number values data Nat : Ty where ze : ze ≈ ze ∈ Nat su : a ≈ b ∈ Nat → ----------------- su a ≈ su b ∈ Nat ne : c ≈ c′ ∈ Bot → -------------------- ↑ N c ≈ ↑ N c′ ∈ Nat -- Neutral type values are related simply when the neutral values themselves are related by Bot data Neu : Ty where ne : c ≈ c′ ∈ Bot → --------------------- ↑ A c ≈ ↑ A′ c′ ∈ Neu -- Now we move on to defining the PER model. To model the universes, we use -- Tarski-style encoding, i.e. for a universe level i, 𝕌 i is a PER relating two -- domain values (or "codes") that are types (i.e. elements of the i'th level -- universe). If A ≈ B ∈ 𝕌 i, then El i A relates two values that are elements of the -- set encoded by A. -- -- Unfortunately, this method only works on paper. In type theory, we must consider -- the effect of proof relevance, when defining El, we must take a witness -- A≈B : A ≈ B ∈ 𝕌 i instead of just A, and El is defined by recursion on A≈B, while 𝕌 -- itself is defined inductively, hence a induction-recursion definition. -- -- Finally, we need a well-founded definition in order to handle cumulative -- universe. In a non-cumulative setting, we expect that this extra well-founded layer -- by the universe level can be taken away, because we can only grow the universe -- level by one each time. -- Helper definitions for the PER model -- The record for relating return types of Π's -- -- Here R is always 𝕌 i, so on paper, it represents ⟦T⟧(ρ) ≈ ⟦T′⟧(ρ′) ∈ 𝕌 i. record ΠRT T ρ T′ ρ′ R : Set where field ⟦T⟧ : D ⟦T′⟧ : D ↘⟦T⟧ : ⟦ T ⟧ ρ ↘ ⟦T⟧ ↘⟦T′⟧ : ⟦ T′ ⟧ ρ′ ↘ ⟦T′⟧ T≈T′ : ⟦T⟧ ≈ ⟦T′⟧ ∈ R -- The record for relating values of type Π A T ρ -- -- Here R is always El i ⟦T⟧(ρ ↦ a), so on paper, it represents f a ≈ f′ a′ ∈ El i ⟦T⟧(ρ ↦ a). record Π̂ (f a f′ a′ : D) R : Set where field fa : D fa′ : D ↘fa : f ∙ a ↘ fa ↘fa′ : f′ ∙ a′ ↘ fa′ fa≈fa′ : fa ≈ fa′ ∈ R module PERDef (i : ℕ) (Univ : ∀ {j} → j < i → Ty) where mutual data 𝕌 : Ty where ne : C ≈ C′ ∈ Bot → ↑ A C ≈ ↑ A′ C′ ∈ 𝕌 N : N ≈ N ∈ 𝕌 U : ∀ {j j′} → j < i → -- cumulativity only requires j < i j ≡ j′ → -- keeping equality here helps with --without-K settings -------------- U j ≈ U j′ ∈ 𝕌 Π : (iA : A ≈ A′ ∈ 𝕌) → (∀ {a a′} → a ≈ a′ ∈ El iA → ΠRT T (ρ ↦ a) T′ (ρ′ ↦ a′) 𝕌) → ------------------------- Π A T ρ ≈ Π A′ T′ ρ′ ∈ 𝕌 El : A ≈ B ∈ 𝕌 → Ty El (ne C≈C′) = Neu El N = Nat El (U j<i eq) = Univ j<i El (Π iA RT) = λ f f′ → ∀ {a b} (inp : a ≈ b ∈ El iA) → Π̂ (f) a (f′) b (El (ΠRT.T≈T′ (RT inp))) -- Now we tie the knot and expose 𝕌 and El in the wild. 𝕌-wellfounded : ∀ i {j} → j < i → Ty 𝕌-wellfounded .(suc _) (s≤s {j} j<i) = PERDef.𝕌 j (λ j′<j → 𝕌-wellfounded _ (≤-trans j′<j j<i)) private module M i = PERDef i (𝕌-wellfounded i) open M hiding (𝕌; El) public pattern U′ i<j = U i<j refl 𝕌 : ℕ → Ty 𝕌 = M.𝕌 -- cannot omit `i`. not sure why El : ∀ i → A ≈ B ∈ 𝕌 i → Ty El i = M.El i -- On paper, it represents ⟦T⟧(ρ) ≈ ⟦T′⟧(ρ′) ∈ 𝕌 i. record RelTyp i T ρ T′ ρ′ : Set where field ⟦T⟧ : D ⟦T′⟧ : D ↘⟦T⟧ : ⟦ T ⟧ ρ ↘ ⟦T⟧ ↘⟦T′⟧ : ⟦ T′ ⟧ ρ′ ↘ ⟦T′⟧ T≈T′ : ⟦T⟧ ≈ ⟦T′⟧ ∈ 𝕌 i -- PER model for context stacks and global evaluation environments -- -- Again we use induction-recursion here in order to model related context stacks and -- related evaluation environments. -- -- ⊨ Γ ≈ Δ means that Γ and Δ are two related contexts so that every -- corresponding types in them are related after evaluation. The PER for -- evaluation environments is defined recursively on ⊨ Γ ≈ Δ. On paper, we write -- -- ρ ≈ ρ′ ∈ ⟦ Γ ⟧ -- -- where ⊨ Γ ≈ Γ, but this again will not work in Agda due to proof relevance. For -- this reason, we must prove some extra properties. infix 4 ⊨_≈_ ⊨_ mutual data ⊨_≈_ : Ctx → Ctx → Set where []-≈ : ⊨ [] ≈ [] ∷-cong : ∀ {i} → (Γ≈Δ : ⊨ Γ ≈ Δ) → (∀ {ρ ρ′} → ρ ≈ ρ′ ∈ ⟦ Γ≈Δ ⟧ρ → RelTyp i T ρ T′ ρ′) → ---------------- ⊨ T ∷ Γ ≈ T′ ∷ Δ ⟦_⟧ρ : ⊨ Γ ≈ Δ → Ev ⟦ []-≈ ⟧ρ ρ ρ′ = ⊤ ⟦ ∷-cong Γ≈Δ rel ⟧ρ ρ ρ′ = Σ (drop ρ ≈ drop ρ′ ∈ ⟦ Γ≈Δ ⟧ρ) λ ρ≈ρ′ → let open RelTyp (rel ρ≈ρ′) in lookup ρ 0 ≈ lookup ρ′ 0 ∈ El _ T≈T′ ⊨_ : Ctx → Set ⊨ Γ = ⊨ Γ ≈ Γ