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Section-3.agda
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Section-3.agda
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module ROmega.Examples.Section-3 where
open import Agda.Primitive
open import Level
open import Data.Product
renaming (proj₁ to fst; proj₂ to snd)
hiding (Σ)
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym)
open import Data.String
open import Data.Unit.Polymorphic
open import Data.Fin renaming (zero to fzero ; suc to fsuc)
open import ROmega.Kinds
open import ROmega.Entailment -- extensionality
open import ROmega.Entailment.Reasoning
open import ROmega.Types hiding (U)
open import ROmega.Types.Substitution
open import ROmega.Types.Substitution.Properties -- extensionality
open import ROmega.Equivalence.Syntax
open import ROmega.Terms -- extensionality
--------------------------------------------------------------------------------
-- We work in the simple row theory.
open SimpleRowSyntax
open SimpleRowSemantics
open TermSyntax Ent
open TermSemantics Ent ⟦_⟧n
--------------------------------------------------------------------------------
-- idω : ★ → ★ at all levels.
idω : ∀ {ℓ ℓΔ} {Δ : KEnv ℓΔ} → Type Δ ((★ ℓ) `→ (★ ℓ))
idω {ℓ} = `λ (★ ℓ) (tvar Z)
-- #############################################################################
-- 3.1 First-Class Labels.
-- #############################################################################
--------------------------------------------------------------------------------
-- Select (sel).
--
-- We have:
-- sel : ∀ Ł : L, T : ★, ζ : R[ ★ ]. (Ł R▹ T) ≲ ζ ⇒ Π ζ → ⌊ Ł ⌋ → T
-- sel = Λ Ł : L. Λ T : ★. Λ ζ : R[ ★ ]. ƛ p : (Ł R▹ T) ≲ ζ ⇒ Π ζ.
-- λ r : Π ζ. λ l : ⌊ Ł ⌋. (prj r) / l
--
--------------------------------------------------------------------------------
selT : ∀ {ℓ ℓΔ} {Δ : KEnv ℓΔ} → Type Δ (★ (lsuc ℓ))
selT {ℓ} =
`∀ (L {lzero}) (`∀ (★ ℓ) (`∀ R[ ★ ℓ ]
((Ł R▹ T) ≲ ζ ⇒ Π ζ `→ ⌊_⌋ {ι = lzero} Ł `→ T)))
where
ζ = tvar Z
T = tvar (S Z)
Ł = tvar (S (S Z))
sel : ∀ {ℓ ℓΔ ℓφ ℓΓ} {Δ : KEnv ℓΔ} {φ : PEnv Δ ℓφ} {Γ : Env Δ ℓΓ} → Term Δ φ Γ (selT {ℓ})
sel {ℓ} = `Λ L (`Λ (★ ℓ) (`Λ R[ (★ ℓ) ]
(`ƛ ((Ł R▹ T) ≲ ζ) (`λ (Π ζ) (`λ ⌊ Ł ⌋ body)))))
where
ζ = tvar Z
T = tvar (S Z)
Ł = tvar (S (S Z))
body = (prj▹ (var (S Z)) (n-var Z)) / (var Z)
--------------------------------------------------------------------------------
-- Construct (con).
conT : ∀ {ℓ ℓΔ} {Δ : KEnv ℓΔ} → Type Δ (★ (lsuc ℓ))
conT {ℓ} =
`∀ (L {lzero}) (`∀ (★ ℓ) (`∀ R[ (★ ℓ) ]
((l R▹ t) ≲ z ⇒ ⌊_⌋ {ι = lzero} l `→ t `→ Σ z)))
where
z = tvar Z
t = tvar (S Z)
l = tvar (S (S Z))
con : ∀ {ℓ ℓΔ ℓφ ℓΓ} {Δ : KEnv ℓΔ} {φ : PEnv Δ ℓφ} {Γ : Env Δ ℓΓ} → Term Δ φ Γ (conT {ℓ})
con {ℓ} = `Λ L (`Λ (★ ℓ) (`Λ R[ (★ ℓ) ]
(`ƛ ((l R▹ t) ≲ z) ((`λ (⌊ l ⌋) (`λ t Σz))))))
where
z = tvar Z
t = tvar (S Z)
l = tvar (S (S Z))
x = var Z
l' = var (S Z)
Σz = inj▹ (l' ▹ x) (n-var Z)
-- Some assertions about con.
con₁ con₂ : ∀ {ℓ} → ⟦ conT {ℓ} ⟧t tt
con₁ _ t z π ρ x with π fzero
... | n , eq rewrite eq = n , x
con₂ = ⟦ con ⟧ tt tt tt
con-ext-eq : ∀ {ℓ} u X z π ρ u' → con₁ {ℓ} u X z π ρ u' ≡ con₂ {ℓ} u X z π ρ u'
con-ext-eq _ X row π r _ with π fzero
... | m , eq rewrite eq = refl
--------------------------------------------------------------------------------
-- Case (case).
caseT : ∀ {ℓ ℓΔ} {Δ : KEnv ℓΔ} → Type Δ (★ (lsuc ℓ))
caseT {ℓ} =
`∀ (L {lzero}) (`∀ (★ ℓ) (`∀ (★ ℓ)
(⌊_⌋ {ι = lzero} l `→ (t `→ u) `→ Σ (l R▹ t) `→ u)))
where
l = tvar (S (S Z))
t = tvar (S Z)
u = tvar Z
case : ∀ {ℓ ℓΔ ℓΦ ℓΓ} {Δ : KEnv ℓΔ} {Φ : PEnv Δ ℓΦ} {Γ : Env Δ ℓΓ} →
Term Δ Φ Γ (caseT {ℓ})
case {ℓ} = `Λ L (`Λ (★ ℓ) (`Λ (★ ℓ)
(`λ ⌊ Ł ⌋ (`λ (T `→ U) (`λ (Σ (Ł R▹ T)) (f · ((Σ⁻¹ x) / l)))))))
where
Ł = tvar (S (S Z))
T = tvar (S Z)
U = tvar Z
l = var (S (S Z))
f = var (S Z)
x = var Z
--------------------------------------------------------------------------------
-- If Then Else (ifte).
Tru Fls : ∀ {ℓΔ} {Δ : KEnv ℓΔ} →
Type Δ (L {lzero})
Tru = lab "True"
Fls = lab "False"
BoolP : ∀ {ℓ ℓΔ} {Δ : KEnv ℓΔ} → Pred (Δ , R[ ★ ℓ ]) (★ ℓ)
BoolP = (Tru R▹ ∅) · Fls R▹ ∅ ~ tvar Z
Bool : ∀ {ℓ} {ℓΔ} {Δ : KEnv ℓΔ} →
Type Δ (★ (lsuc ℓ))
Bool {ℓ} = `∀ (R[ ★ ℓ ]) (BoolP ⇒ Σ (tvar Z))
ifteT : ∀ {ℓ} {ℓΔ} {Δ : KEnv ℓΔ} →
Type Δ (★ (lsuc ℓ))
ifteT {ℓ} = `∀ (★ ℓ) (`∀ R[ ★ ℓ ]
(BoolP {ℓ} ⇒ Bool {ℓ} `→ tvar (S Z) `→ tvar (S Z) `→ tvar (S Z)))
ifte : ∀ {ℓ ℓΔ ℓφ ℓΓ} {Δ : KEnv ℓΔ} {φ : PEnv Δ ℓφ} {Γ : Env Δ ℓΓ} →
Term Δ φ Γ (ifteT {ℓ})
ifte =
`Λ (★ _)
(`Λ R[ ★ _ ]
(`ƛ _
(`λ Bool
(`λ (tvar (S Z))
(`λ (tvar (S Z))
((((((((case ·[ Tru ]) ·[ ∅ ]) ·[ _ ]) · lab Tru) · `λ _ (var (S (S Z)))))
▿
((((((case ·[ Fls ]) ·[ ∅ ]) ·[ _ ]) · lab Fls) · `λ _ (var (S Z)))))
(n-var Z) · (((var (S (S Z))) ·[ tvar Z ]) ·⟨ n-var Z ⟩) ))))))
-- #############################################################################
-- 3.2 The Duality of Records.
-- #############################################################################
--------------------------------------------------------------------------------
-- Reification (reify).
reifyT : Type ε ★₁
reifyT = `∀ R[ ★₀ ] (`∀ ★₀ (((Σ z) `→ t) `→ Π (⌈ (`λ ★₀ ((tvar Z) `→ (tvar (S Z)))) ⌉· z)))
where
t = tvar Z
z = tvar (S Z)
reify : Term ε ε ε reifyT
reify = `Λ R[ ★₀ ] (`Λ ★₀ (`λ (((Σ z) `→ t)) (syn z (`λ ★₀ ((tvar Z) `→ (tvar (S Z)))) sbod)))
where
t = tvar Z
z = tvar (S Z)
sbod = `Λ (L {lzero}) (`Λ ★₀ (`Λ R[ ★₀ ] (`ƛ ((l R▹ u) · y ~ z')
(`λ ⌊ l ⌋
(t-≡
(`λ u
(f ·
((((((con ·[ l ]) ·[ u ]) ·[ z' ])
·⟨ n-·≲L (n-var Z) ⟩)
· (var (S Z)))
· (var Z))))
(teq-sym teq-β))))))
where
y = tvar Z
u = tvar (S Z)
l = tvar (S (S Z))
z' = tvar (S (S (S (S Z))))
f = var (S (S Z))
⟦reify⟧ : ⟦ reifyT ⟧t tt
⟦reify⟧ = ⟦ reify ⟧ tt tt tt
--------------------------------------------------------------------------------
-- Reflection (reflect).
-- We have:
-- ana : (ρ : Row κ) (φ : κ → ★) (T : ★) →
-- (∀ Ł : L, U : κ, Y : R[ κ ]. (Ł R▹ U) · Y ~ ρ ⇒ ⌊ Ł ⌋ → φ U → T) →
-- Σ ρ → T
--
-- reflect : ∀ ζ : R[ ★ ], T : ★.
-- Π (lift₂ (λ (X : ★). X → T) ζ) →
-- Σ ζ → T
-- reflect = Λ ζ : R[ ★ ]. Λ T : ★.
-- λ r : Π (lift₂ (λ (X : ★). X → T) ζ).
-- ana ★ ζ (λ X : ★. X) T (
-- Λ Ł : L. Λ U : ★. Λ Y : R[ κ ].
-- ƛ p : (Ł R▹ U) · Y ~ ρ.
-- λ l : ⌊ Ł ⌋. λ u : (λ X. X) U.
-- sel ·[ Ł ] ·[ ((λ X. X) U) → T ] ·[ lift₂ (λ (X : ★). X → T) ζ ]
-- ·⟨ ? ⟩ ·( r ) ·( l ) ·( u ))
reflectT : Type ε ★₁
reflectT = `∀ R[ ★₀ ] (`∀ ★₀
(Π (⌈ (`λ ★₀ ((tvar Z) `→ (tvar (S Z)))) ⌉· z) `→
((Σ (⌈ idω ⌉· z)) `→ t)))
where
t = tvar Z
z = tvar (S Z)
reflect : Term ε ε ε reflectT
reflect =
-- Λ ζ : R[ ★ ].
`Λ R[ ★₀ ]
-- Λ T : ★.
(`Λ ★₀
-- λ r : Π (ζ → T).
(`λ (Π (⌈ (`λ ★₀ (tvar Z `→ tvar (S Z))) ⌉· (tvar (S Z))))
(ana (tvar (S Z)) idω (tvar Z) M)))
where
M =
-- Λ Ł : L.
`Λ L
-- Λ U : ★
(`Λ ★₀
-- Λ Y : R[ κ ].
(`Λ R[ ★₀ ]
-- ƛ p : (Ł R▹ U) · Y ~ Ζ
(`ƛ ((tvar (S (S Z)) R▹ tvar (S Z)) · (tvar Z) ~ (tvar (S (S (S (S Z))))))
-- λ l : ⌊ Ł ⌋
(`λ ⌊ (tvar (S (S Z))) ⌋
-- λ u : (idω ·U).
(`λ (idω ·[ tvar (S Z) ])
-- (sel body)
body)))))
where
body =
((((((sel
-- ·[ Ł ]
·[ tvar (S (S Z)) ])
-- ·[ ((λ X. X) U) → T ]
·[ idω ·[ tvar (S Z) ] `→ (tvar (S (S (S Z)))) ])
-- ·[ lift₂ (λ (X : ★). X → T) ζ ]
·[ (⌈ (`λ ★₀ (tvar Z `→ tvar (S (S (S (S Z)))))) ⌉· (tvar (S (S (S (S Z)))))) ])
-- ·⟨ evidence ⟩
·⟨ evidence ⟩)
-- · r
· var (S (S Z)))
-- · l
· (var (S Z )))
-- · u
· (var Z)
where
Ł = tvar (S (S Z))
T = (tvar (S (S (S Z))))
T' = (tvar (S (S (S (S Z)))))
Uu = tvar (S Z)
Y = tvar Z
ζ = tvar (S (S (S (S Z))))
evidence : Ent _ _ ((Ł R▹ ((idω ·[ Uu ]) `→ T)) ≲ ⌈ (`λ ★₀ (tvar Z `→ T')) ⌉· ζ)
evidence =
(((Ł R▹ Uu) · Y ~ ζ)
⊩⟨ n-·≲L ⟩
((Ł R▹ Uu) ≲ ζ)
⊩⟨ n-≡ (peq-≲ (teq-sing teq-refl (teq-sym teq-β)) teq-refl) ⟩
((Ł R▹ (idω ·[ Uu ])) ≲ ζ)
⊩⟨ n-≲lift₂ ⟩
((⌈ (`λ ★₀ (tvar Z `→ T')) ⌉· (Ł R▹ (idω ·[ Uu ])) ≲ ⌈ (`λ ★₀ (tvar Z `→ T')) ⌉· ζ))
⊩⟨ n-≡ (peq-≲ teq-lift₂ teq-refl) ⟩
(((Ł R▹ ((`λ ★₀ (tvar Z `→ T')) ·[ (idω ·[ Uu ]) ])) ≲ ⌈ (`λ ★₀ (tvar Z `→ T')) ⌉· ζ))
⊩⟨ n-≡ (peq-≲ (teq-sing teq-refl teq-β) teq-refl) ⟩
∎)
(n-var Z)
-- #############################################################################
-- 3.3 Transformations.
-- #############################################################################
--------------------------------------------------------------------------------
-- Iterators (Iter).
Iter : ∀ {Δ : KEnv lzero} →
(κ : Kind lzero) →
Type Δ (κ `→ ★₀) →
Type Δ (κ `→ ★₀) →
Type Δ (R[ κ ]) →
Type Δ ★₁
Iter κ f g z =
`∀ (L {lzero}) (`∀ κ (`∀ R[ κ ]
((Ł R▹ U) · Y ~ z' ⇒ ⌊_⌋ {ι = lzero} Ł `→ f' ·[ U ] `→ g' ·[ U ])))
where
z' = weaken (weaken (weaken z))
f' = weaken (weaken (weaken f))
g' = weaken (weaken (weaken g))
Ł = tvar (S (S Z))
U = tvar (S Z)
Y = tvar Z
--------------------------------------------------------------------------------
-- mapping over records (map-π).
map-ΠT : ∀ {Δ : KEnv lzero} →
(κ : Kind lzero) →
Type Δ ★₁
map-ΠT κ =
`∀ R[ κ ] (`∀ (κ `→ ★₀) (`∀ (κ `→ ★₀)
(Iter κ f g z `→ (Π (⌈ f ⌉· z)) `→ Π (⌈ g ⌉· z) )))
where
g = tvar Z
f = tvar (S Z)
z = tvar (S (S Z))
map-Π : ∀ {Δ : KEnv lzero} {Φ : PEnv Δ lzero} {Γ : Env Δ lzero} →
(κ : Kind lzero) →
Term Δ Φ Γ (map-ΠT κ)
map-Π κ =
`Λ {- z -} R[ κ ]
(`Λ {- f -} (κ `→ ★₀)
(`Λ {- g -} (κ `→ ★₀)
(`λ {- i -} (Iter κ (tvar (S Z)) (tvar Z) (tvar (S (S Z))))
(`λ {- r -} (Π (⌈ f ⌉· tvar (S (S Z))))
(syn (tvar (S (S Z))) (tvar Z)
(`Λ {- Ł -} L
(`Λ {- U -} κ
(`Λ {- Y -} R[ κ ]
(`ƛ {- _ -} ((tvar (S (S Z)) R▹ tvar (S Z)) · (tvar Z) ~ tvar (S (S (S (S (S Z))))))
(`λ {- l -} ⌊ tvar (S (S Z)) ⌋
((i' · l) · ((sel' · r) · l)))
)))))))))
where
f = tvar (S Z)
l = var Z
r = var (S Z)
i' = let
Ł = tvar (S (S Z))
U = tvar (S Z)
Y = tvar Z
in ((((var (S (S Z))) ·[ Ł ]) ·[ U ]) ·[ Y ]) ·⟨ n-var Z ⟩
sel' = let
Ł = tvar (S (S Z))
U = tvar (S Z)
-- Y = tvar Z
z' = tvar (S (S (S (S (S Z)))))
f' = tvar (S (S (S (S Z))))
in
(((sel
·[ Ł ])
·[ f' ·[ U ] ])
·[ (⌈ f' ⌉· z') ])
·⟨ evidence ⟩
where
evidence : let
Ł = tvar (S (S Z))
U = tvar (S Z)
z' = tvar (S (S (S (S (S Z)))))
f' = tvar (S (S (S (S Z))))
in Ent _ _ ((Ł R▹ (f' ·[ U ]) ) ≲ (⌈ f' ⌉· z'))
evidence = let
Ł = tvar (S (S Z))
U = tvar (S Z)
Y = tvar Z
z' = tvar (S (S (S (S (S Z)))))
f' = tvar (S (S (S (S Z))))
in
((((Ł R▹ U) · Y ~ z')
⊩⟨ n-·≲L ⟩
((Ł R▹ U) ≲ z'
⊩⟨ n-≲lift₂ ⟩
⌈ f' ⌉· (Ł R▹ U ) ≲ ⌈ f' ⌉· z'
⊩⟨ n-≡ (peq-≲ teq-lift₂ teq-refl) ⟩
∎)) (n-var Z))
--------------------------------------------------------------------------------
-- Mapping over Variants.
map-ΣT : ∀ {Δ : KEnv lzero} →
(κ : Kind lzero) →
Type Δ ★₁
map-ΣT κ =
`∀ R[ κ ] (`∀ (κ `→ ★₀) (`∀ (κ `→ ★₀)
(Iter κ f g z `→ (Σ (⌈ f ⌉· z)) `→ Σ (⌈ g ⌉· z) )))
where
g = tvar Z
f = tvar (S Z)
z = tvar (S (S Z))
map-Σ : ∀ {Δ : KEnv lzero} {Φ : PEnv Δ lzero} {Γ : Env Δ lzero} →
(κ : Kind lzero) →
Term Δ Φ Γ (map-ΣT κ)
map-Σ κ =
`Λ {- z -} R[ κ ]
(`Λ {- f -} (κ `→ ★₀)
(`Λ {- g -} (κ `→ ★₀)
(`λ {- i -} (Iter κ (tvar (S Z)) (tvar Z) (tvar (S (S Z))))
(`λ {- v -} (Σ (⌈ f ⌉· tvar (S (S Z))))
((ana (tvar (S (S Z))) (tvar (S Z)) (Σ (⌈ tvar Z ⌉· (tvar (S (S Z)))))
(`Λ {- Ł -} L
(`Λ {- U -} κ
(`Λ {- Y -} R[ κ ]
(`ƛ {- _ -} ((tvar (S (S Z)) R▹ tvar (S Z)) · (tvar Z) ~ tvar (S (S (S (S (S Z))))))
(`λ {- l -} ⌊ tvar (S (S Z)) ⌋
(`λ {- x -} (tvar (S (S (S (S Z)))) ·[ (tvar (S Z)) ])
(((con' · l) · ((i' · l) · x)
))))))))) · (var Z))))))
where
f = tvar (S Z)
x = var Z
l = var (S Z)
i' =
let
Ł = tvar (S (S Z))
U = tvar (S Z)
Y = tvar Z
in ((((var (S (S (S Z)))) ·[ Ł ]) ·[ U ]) ·[ Y ]) ·⟨ n-var Z ⟩
con' =
let
Ł = tvar (S (S Z))
U = tvar (S Z)
-- Y = tvar Z
z' = tvar (S (S (S (S (S Z)))))
g' = tvar (S (S (S Z)))
in (((con
·[ Ł ])
·[ g' ·[ U ] ])
·[ ⌈ g' ⌉· z' ])
·⟨ evidence ⟩
where
evidence : let
Ł = tvar (S (S Z))
U = tvar (S Z)
z' = tvar (S (S (S (S (S Z)))))
g' = tvar (S (S (S Z)))
in Ent _ _ ((Ł R▹ (g' ·[ U ]) ) ≲ (⌈ g' ⌉· z'))
evidence = let
Ł = tvar (S (S Z))
U = tvar (S Z)
Y = tvar Z
z' = tvar (S (S (S (S (S Z)))))
g' = tvar (S (S (S Z)))
in
((((Ł R▹ U) · Y ~ z')
⊩⟨ n-·≲L ⟩
((Ł R▹ U) ≲ z'
⊩⟨ n-≲lift₂ ⟩
⌈ g' ⌉· (Ł R▹ U ) ≲ ⌈ g' ⌉· z'
⊩⟨ n-≡ (peq-≲ teq-lift₂ teq-refl) ⟩
∎)) (n-var Z))