{-# OPTIONS --rewriting #-}
module A201801.SequentCalculusDraft where
open import A201801.Prelude
open import A201801.Category
open import A201801.List
open import A201801.ListLemmas
open import A201801.AllList
open import A201801.FullIPLPropositions
open import A201801.FullIPLDerivations hiding (cut)
infix 4 _⊒_
_⊒_ : ∀ {X} → List X → List X → Set
Ξ′ ⊒ Ξ = ∀ {A} → Ξ ∋ A → Ξ′ ∋ A
bot⊒ : ∀ {X} → {Ξ : List X}
→ Ξ ⊒ ∙
bot⊒ {Ξ = ∙} ()
bot⊒ {Ξ = Ξ , A} i = suc (bot⊒ i)
keep⊒ : ∀ {X A} → {Ξ Ξ′ : List X}
→ Ξ′ ⊒ Ξ
→ Ξ′ , A ⊒ Ξ , A
keep⊒ η zero = zero
keep⊒ η (suc i) = suc (η i)
ex⊒ : ∀ {X A B} → {Ξ : List X}
→ (Ξ , B) , A ⊒ (Ξ , A) , B
ex⊒ zero = suc zero
ex⊒ (suc zero) = zero
ex⊒ (suc (suc i)) = suc (suc i)
ct⊒ : ∀ {X A} → {Ξ : List X}
→ Ξ , A ⊒ (Ξ , A) , A
ct⊒ zero = zero
ct⊒ (suc zero) = zero
ct⊒ (suc (suc i)) = suc i
unct⊒ : ∀ {X A} → {Ξ : List X}
→ (Ξ , A) , A ⊒ Ξ , A
unct⊒ zero = zero
unct⊒ (suc i) = suc (suc i)
wk⊒ : ∀ {X A} → {Ξ : List X}
→ Ξ , A ⊒ Ξ
wk⊒ i = suc i
genct⊒ : ∀ {X A} → {Ξ : List X}
→ Ξ ∋ A
→ Ξ ⊒ Ξ , A
genct⊒ i zero = i
genct⊒ i (suc j) = j
_-_ : ∀ {X A} → (Ξ : List X) → Ξ ∋ A → List X
∙ - ()
(Ξ , A) - zero = Ξ
(Ξ , B) - suc i = (Ξ - i) , B
del⊇ : ∀ {X A} → {Ξ : List X}
→ (i : Ξ ∋ A)
→ Ξ ⊇ Ξ - i
del⊇ zero = drop id
del⊇ (suc i) = keep (del⊇ i)
data _≡∋_ {X} : ∀ {A B} → {Ξ : List X} → Ξ ∋ A → Ξ ∋ B → Set
where
same : ∀ {A} → {Ξ : List X}
→ (i : Ξ ∋ A)
→ i ≡∋ i
diff : ∀ {A B} → {Ξ : List X}
→ (i : Ξ ∋ A) (j : Ξ - i ∋ B)
→ i ≡∋ ren∋ (del⊇ i) j
_≟∋_ : ∀ {X A B} → {Ξ : List X}
→ (i : Ξ ∋ A) (j : Ξ ∋ B)
→ i ≡∋ j
zero ≟∋ zero = same zero
zero ≟∋ suc j rewrite id-ren∋ j ⁻¹ = diff zero j
suc i ≟∋ zero = diff (suc i) zero
suc i ≟∋ suc j with i ≟∋ j
suc i ≟∋ suc .i | same .i = same (suc i)
suc i ≟∋ suc ._ | diff .i j = diff (suc i) (suc j)
delred⊒ : ∀ {X A} → {Ξ : List X}
→ (i : Ξ ∋ A) (j : Ξ - i ∋ A)
→ Ξ - i ⊒ Ξ
delred⊒ i j k with i ≟∋ k
delred⊒ i j k | same .k = j
delred⊒ i j _ | diff .i k = k
del⊒ : ∀ {X A} → {Ξ : List X}
→ (i : Ξ ∋ A)
→ Ξ ⊒ Ξ - i
del⊒ zero j = suc j
del⊒ (suc i) j = keep⊒ (del⊒ i) j
mutual
infix 3 _⊢_normal
data _⊢_normal : List Form → Form → Set
where
lam : ∀ {A B Γ} → Γ , A ⊢ B normal
→ Γ ⊢ A ⊃ B normal
pair : ∀ {A B Γ} → Γ ⊢ A normal → Γ ⊢ B normal
→ Γ ⊢ A ∧ B normal
unit : ∀ {Γ} → Γ ⊢ 𝟏 normal
abort : ∀ {A Γ} → Γ ⊢ 𝟎 neutral
→ Γ ⊢ A normal
inl : ∀ {A B Γ} → Γ ⊢ A normal
→ Γ ⊢ A ∨ B normal
inr : ∀ {A B Γ} → Γ ⊢ B normal
→ Γ ⊢ A ∨ B normal
case : ∀ {A B C Γ} → Γ ⊢ A ∨ B neutral → Γ , A ⊢ C normal → Γ , B ⊢ C normal
→ Γ ⊢ C normal
ent : ∀ {A Γ} → Γ ⊢ A neutral
→ Γ ⊢ A normal
infix 3 _⊢_neutral
data _⊢_neutral : List Form → Form → Set
where
var : ∀ {A Γ} → Γ ∋ A
→ Γ ⊢ A neutral
app : ∀ {A B Γ} → Γ ⊢ A ⊃ B neutral → Γ ⊢ A normal
→ Γ ⊢ B neutral
fst : ∀ {A B Γ} → Γ ⊢ A ∧ B neutral
→ Γ ⊢ A neutral
snd : ∀ {A B Γ} → Γ ⊢ A ∧ B neutral
→ Γ ⊢ B neutral
infix 3 _⊢_allneutral
_⊢_allneutral : List Form → List Form → Set
Γ ⊢ Ξ allneutral = All (Γ ⊢_neutral) Ξ
infix 3 _⊢_allnormal
_⊢_allnormal : List Form → List Form → Set
Γ ⊢ Ξ allnormal = All (Γ ⊢_normal) Ξ
mutual
renₙₘ : ∀ {Γ Γ′ A} → Γ′ ⊇ Γ → Γ ⊢ A normal
→ Γ′ ⊢ A normal
renₙₘ η (lam 𝒟) = lam (renₙₘ (keep η) 𝒟)
renₙₘ η (pair 𝒟 ℰ) = pair (renₙₘ η 𝒟) (renₙₘ η ℰ)
renₙₘ η unit = unit
renₙₘ η (abort 𝒟) = abort (renₙₜ η 𝒟)
renₙₘ η (inl 𝒟) = inl (renₙₘ η 𝒟)
renₙₘ η (inr 𝒟) = inr (renₙₘ η 𝒟)
renₙₘ η (case 𝒟 ℰ ℱ) = case (renₙₜ η 𝒟) (renₙₘ (keep η) ℰ) (renₙₘ (keep η) ℱ)
renₙₘ η (ent 𝒟) = ent (renₙₜ η 𝒟)
renₙₜ : ∀ {Γ Γ′ A} → Γ′ ⊇ Γ → Γ ⊢ A neutral
→ Γ′ ⊢ A neutral
renₙₜ η (var i) = var (ren∋ η i)
renₙₜ η (app 𝒟 ℰ) = app (renₙₜ η 𝒟) (renₙₘ η ℰ)
renₙₜ η (fst 𝒟) = fst (renₙₜ η 𝒟)
renₙₜ η (snd 𝒟) = snd (renₙₜ η 𝒟)
rensₙₜ : ∀ {Γ Γ′ Ξ} → Γ′ ⊇ Γ → Γ ⊢ Ξ allneutral
→ Γ′ ⊢ Ξ allneutral
rensₙₜ η ξ = maps (renₙₜ η) ξ
wkₙₜ : ∀ {B Γ A} → Γ ⊢ A neutral
→ Γ , B ⊢ A neutral
wkₙₜ 𝒟 = renₙₜ (drop id) 𝒟
wksₙₜ : ∀ {A Γ Ξ} → Γ ⊢ Ξ allneutral
→ Γ , A ⊢ Ξ allneutral
wksₙₜ ξ = rensₙₜ (drop id) ξ
vzₙₜ : ∀ {Γ A} → Γ , A ⊢ A neutral
vzₙₜ = var zero
liftsₙₜ : ∀ {A Γ Ξ} → Γ ⊢ Ξ allneutral
→ Γ , A ⊢ Ξ , A allneutral
liftsₙₜ ξ = wksₙₜ ξ , vzₙₜ
varsₙₜ : ∀ {Γ Γ′} → Γ′ ⊇ Γ
→ Γ′ ⊢ Γ allneutral
varsₙₜ done = ∙
varsₙₜ (drop η) = wksₙₜ (varsₙₜ η)
varsₙₜ (keep η) = liftsₙₜ (varsₙₜ η)
idsₙₜ : ∀ {Γ} → Γ ⊢ Γ allneutral
idsₙₜ = varsₙₜ id
rensₙₘ : ∀ {Γ Γ′ Ξ} → Γ′ ⊇ Γ → Γ ⊢ Ξ allnormal
→ Γ′ ⊢ Ξ allnormal
rensₙₘ η ξ = maps (renₙₘ η) ξ
wkₙₘ : ∀ {B Γ A} → Γ ⊢ A normal
→ Γ , B ⊢ A normal
wkₙₘ 𝒟 = renₙₘ (drop id) 𝒟
wksₙₘ : ∀ {A Γ Ξ} → Γ ⊢ Ξ allnormal
→ Γ , A ⊢ Ξ allnormal
wksₙₘ ξ = rensₙₘ (drop id) ξ
vzₙₘ : ∀ {Γ A} → Γ , A ⊢ A normal
vzₙₘ = ent vzₙₜ
liftsₙₘ : ∀ {A Γ Ξ} → Γ ⊢ Ξ allnormal
→ Γ , A ⊢ Ξ , A allnormal
liftsₙₘ ξ = wksₙₘ ξ , vzₙₘ
varsₙₘ : ∀ {Γ Γ′} → Γ′ ⊇ Γ
→ Γ′ ⊢ Γ allnormal
varsₙₘ done = ∙
varsₙₘ (drop η) = wksₙₘ (varsₙₘ η)
varsₙₘ (keep η) = liftsₙₘ (varsₙₘ η)
idsₙₘ : ∀ {Γ} → Γ ⊢ Γ allnormal
idsₙₘ = varsₙₘ id
mutual
subₙₘ : ∀ {Γ Ξ A} → Γ ⊢ Ξ allneutral → Ξ ⊢ A normal
→ Γ ⊢ A normal
subₙₘ ξ (lam 𝒟) = lam (subₙₘ (liftsₙₜ ξ) 𝒟)
subₙₘ ξ (pair 𝒟 ℰ) = pair (subₙₘ ξ 𝒟) (subₙₘ ξ ℰ)
subₙₘ ξ unit = unit
subₙₘ ξ (abort 𝒟) = abort (subₙₜ ξ 𝒟)
subₙₘ ξ (inl 𝒟) = inl (subₙₘ ξ 𝒟)
subₙₘ ξ (inr 𝒟) = inr (subₙₘ ξ 𝒟)
subₙₘ ξ (case 𝒟 ℰ ℱ) = case (subₙₜ ξ 𝒟) (subₙₘ (liftsₙₜ ξ) ℰ) (subₙₘ (liftsₙₜ ξ) ℱ)
subₙₘ ξ (ent 𝒟) = ent (subₙₜ ξ 𝒟)
subₙₜ : ∀ {Γ Ξ A} → Γ ⊢ Ξ allneutral → Ξ ⊢ A neutral
→ Γ ⊢ A neutral
subₙₜ ξ (var i) = get ξ i
subₙₜ ξ (app 𝒟 ℰ) = app (subₙₜ ξ 𝒟) (subₙₘ ξ ℰ)
subₙₜ ξ (fst 𝒟) = fst (subₙₜ ξ 𝒟)
subₙₜ ξ (snd 𝒟) = snd (subₙₜ ξ 𝒟)
cutₙₘ : ∀ {Γ A B} → Γ ⊢ A neutral → Γ , A ⊢ B normal
→ Γ ⊢ B normal
cutₙₘ 𝒟 ℰ = subₙₘ (idsₙₜ , 𝒟) ℰ
mutual
thm31ₙₘ : ∀ {Γ A} → Γ ⊢ A normal
→ Γ ⊢ A true
thm31ₙₘ (lam 𝒟) = lam (thm31ₙₘ 𝒟)
thm31ₙₘ (pair 𝒟 ℰ) = pair (thm31ₙₘ 𝒟) (thm31ₙₘ ℰ)
thm31ₙₘ unit = unit
thm31ₙₘ (abort 𝒟) = abort (thm31ₙₜ 𝒟)
thm31ₙₘ (inl 𝒟) = inl (thm31ₙₘ 𝒟)
thm31ₙₘ (inr 𝒟) = inr (thm31ₙₘ 𝒟)
thm31ₙₘ (case 𝒟 ℰ ℱ) = case (thm31ₙₜ 𝒟) (thm31ₙₘ ℰ) (thm31ₙₘ ℱ)
thm31ₙₘ (ent 𝒟) = thm31ₙₜ 𝒟
thm31ₙₜ : ∀ {Γ A} → Γ ⊢ A neutral
→ Γ ⊢ A true
thm31ₙₜ (var i) = var i
thm31ₙₜ (app 𝒟 ℰ) = app (thm31ₙₜ 𝒟) (thm31ₙₘ ℰ)
thm31ₙₜ (fst 𝒟) = fst (thm31ₙₜ 𝒟)
thm31ₙₜ (snd 𝒟) = snd (thm31ₙₜ 𝒟)
mutual
infix 3 _⊢₊_normal
data _⊢₊_normal : List Form → Form → Set
where
lam : ∀ {A B Γ} → Γ , A ⊢₊ B normal
→ Γ ⊢₊ A ⊃ B normal
pair : ∀ {A B Γ} → Γ ⊢₊ A normal → Γ ⊢₊ B normal
→ Γ ⊢₊ A ∧ B normal
unit : ∀ {Γ} → Γ ⊢₊ 𝟏 normal
abort : ∀ {A Γ} → Γ ⊢₊ 𝟎 neutral
→ Γ ⊢₊ A normal
inl : ∀ {A B Γ} → Γ ⊢₊ A normal
→ Γ ⊢₊ A ∨ B normal
inr : ∀ {A B Γ} → Γ ⊢₊ B normal
→ Γ ⊢₊ A ∨ B normal
case : ∀ {A B C Γ} → Γ ⊢₊ A ∨ B neutral → Γ , A ⊢₊ C normal → Γ , B ⊢₊ C normal
→ Γ ⊢₊ C normal
ent : ∀ {A Γ} → Γ ⊢₊ A neutral
→ Γ ⊢₊ A normal
infix 3 _⊢₊_neutral
data _⊢₊_neutral : List Form → Form → Set
where
var : ∀ {A Γ} → Γ ∋ A
→ Γ ⊢₊ A neutral
app : ∀ {A B Γ} → Γ ⊢₊ A ⊃ B neutral → Γ ⊢₊ A normal
→ Γ ⊢₊ B neutral
fst : ∀ {A B Γ} → Γ ⊢₊ A ∧ B neutral
→ Γ ⊢₊ A neutral
snd : ∀ {A B Γ} → Γ ⊢₊ A ∧ B neutral
→ Γ ⊢₊ B neutral
enm : ∀ {A Γ} → Γ ⊢₊ A normal
→ Γ ⊢₊ A neutral
infix 3 _⊢₊_allneutral
_⊢₊_allneutral : List Form → List Form → Set
Γ ⊢₊ Ξ allneutral = All (Γ ⊢₊_neutral) Ξ
infix 3 _⊢₊_allnormal
_⊢₊_allnormal : List Form → List Form → Set
Γ ⊢₊ Ξ allnormal = All (Γ ⊢₊_normal) Ξ
mutual
renₙₘ₊ : ∀ {Γ Γ′ A} → Γ′ ⊇ Γ → Γ ⊢₊ A normal
→ Γ′ ⊢₊ A normal
renₙₘ₊ η (lam 𝒟) = lam (renₙₘ₊ (keep η) 𝒟)
renₙₘ₊ η (pair 𝒟 ℰ) = pair (renₙₘ₊ η 𝒟) (renₙₘ₊ η ℰ)
renₙₘ₊ η unit = unit
renₙₘ₊ η (abort 𝒟) = abort (renₙₜ₊ η 𝒟)
renₙₘ₊ η (inl 𝒟) = inl (renₙₘ₊ η 𝒟)
renₙₘ₊ η (inr 𝒟) = inr (renₙₘ₊ η 𝒟)
renₙₘ₊ η (case 𝒟 ℰ ℱ) = case (renₙₜ₊ η 𝒟) (renₙₘ₊ (keep η) ℰ) (renₙₘ₊ (keep η) ℱ)
renₙₘ₊ η (ent 𝒟) = ent (renₙₜ₊ η 𝒟)
renₙₜ₊ : ∀ {Γ Γ′ A} → Γ′ ⊇ Γ → Γ ⊢₊ A neutral
→ Γ′ ⊢₊ A neutral
renₙₜ₊ η (var i) = var (ren∋ η i)
renₙₜ₊ η (app 𝒟 ℰ) = app (renₙₜ₊ η 𝒟) (renₙₘ₊ η ℰ)
renₙₜ₊ η (fst 𝒟) = fst (renₙₜ₊ η 𝒟)
renₙₜ₊ η (snd 𝒟) = snd (renₙₜ₊ η 𝒟)
renₙₜ₊ η (enm 𝒟) = enm (renₙₘ₊ η 𝒟)
rensₙₜ₊ : ∀ {Γ Γ′ Ξ} → Γ′ ⊇ Γ → Γ ⊢₊ Ξ allneutral
→ Γ′ ⊢₊ Ξ allneutral
rensₙₜ₊ η ξ = maps (renₙₜ₊ η) ξ
wkₙₜ₊ : ∀ {B Γ A} → Γ ⊢₊ A neutral
→ Γ , B ⊢₊ A neutral
wkₙₜ₊ 𝒟 = renₙₜ₊ (drop id) 𝒟
wksₙₜ₊ : ∀ {A Γ Ξ} → Γ ⊢₊ Ξ allneutral
→ Γ , A ⊢₊ Ξ allneutral
wksₙₜ₊ ξ = rensₙₜ₊ (drop id) ξ
vzₙₜ₊ : ∀ {Γ A} → Γ , A ⊢₊ A neutral
vzₙₜ₊ = var zero
liftsₙₜ₊ : ∀ {A Γ Ξ} → Γ ⊢₊ Ξ allneutral
→ Γ , A ⊢₊ Ξ , A allneutral
liftsₙₜ₊ ξ = wksₙₜ₊ ξ , vzₙₜ₊
varsₙₜ₊ : ∀ {Γ Γ′} → Γ′ ⊇ Γ
→ Γ′ ⊢₊ Γ allneutral
varsₙₜ₊ done = ∙
varsₙₜ₊ (drop η) = wksₙₜ₊ (varsₙₜ₊ η)
varsₙₜ₊ (keep η) = liftsₙₜ₊ (varsₙₜ₊ η)
idsₙₜ₊ : ∀ {Γ} → Γ ⊢₊ Γ allneutral
idsₙₜ₊ = varsₙₜ₊ id
rensₙₘ₊ : ∀ {Γ Γ′ Ξ} → Γ′ ⊇ Γ → Γ ⊢₊ Ξ allnormal
→ Γ′ ⊢₊ Ξ allnormal
rensₙₘ₊ η ξ = maps (renₙₘ₊ η) ξ
wkₙₘ₊ : ∀ {B Γ A} → Γ ⊢₊ A normal
→ Γ , B ⊢₊ A normal
wkₙₘ₊ 𝒟 = renₙₘ₊ (drop id) 𝒟
wksₙₘ₊ : ∀ {A Γ Ξ} → Γ ⊢₊ Ξ allnormal
→ Γ , A ⊢₊ Ξ allnormal
wksₙₘ₊ ξ = rensₙₘ₊ (drop id) ξ
vzₙₘ₊ : ∀ {Γ A} → Γ , A ⊢₊ A normal
vzₙₘ₊ = ent vzₙₜ₊
liftsₙₘ₊ : ∀ {A Γ Ξ} → Γ ⊢₊ Ξ allnormal
→ Γ , A ⊢₊ Ξ , A allnormal
liftsₙₘ₊ ξ = wksₙₘ₊ ξ , vzₙₘ₊
varsₙₘ₊ : ∀ {Γ Γ′} → Γ′ ⊇ Γ
→ Γ′ ⊢₊ Γ allnormal
varsₙₘ₊ done = ∙
varsₙₘ₊ (drop η) = wksₙₘ₊ (varsₙₘ₊ η)
varsₙₘ₊ (keep η) = liftsₙₘ₊ (varsₙₘ₊ η)
idsₙₘ₊ : ∀ {Γ} → Γ ⊢₊ Γ allnormal
idsₙₘ₊ = varsₙₘ₊ id
mutual
subₙₘ₊ : ∀ {Γ Ξ A} → Γ ⊢₊ Ξ allneutral → Ξ ⊢₊ A normal
→ Γ ⊢₊ A normal
subₙₘ₊ ξ (lam 𝒟) = lam (subₙₘ₊ (liftsₙₜ₊ ξ) 𝒟)
subₙₘ₊ ξ (pair 𝒟 ℰ) = pair (subₙₘ₊ ξ 𝒟) (subₙₘ₊ ξ ℰ)
subₙₘ₊ ξ unit = unit
subₙₘ₊ ξ (abort 𝒟) = abort (subₙₜ₊ ξ 𝒟)
subₙₘ₊ ξ (inl 𝒟) = inl (subₙₘ₊ ξ 𝒟)
subₙₘ₊ ξ (inr 𝒟) = inr (subₙₘ₊ ξ 𝒟)
subₙₘ₊ ξ (case 𝒟 ℰ ℱ) = case (subₙₜ₊ ξ 𝒟) (subₙₘ₊ (liftsₙₜ₊ ξ) ℰ) (subₙₘ₊ (liftsₙₜ₊ ξ) ℱ)
subₙₘ₊ ξ (ent 𝒟) = ent (subₙₜ₊ ξ 𝒟)
subₙₜ₊ : ∀ {Γ Ξ A} → Γ ⊢₊ Ξ allneutral → Ξ ⊢₊ A neutral
→ Γ ⊢₊ A neutral
subₙₜ₊ ξ (var i) = get ξ i
subₙₜ₊ ξ (app 𝒟 ℰ) = app (subₙₜ₊ ξ 𝒟) (subₙₘ₊ ξ ℰ)
subₙₜ₊ ξ (fst 𝒟) = fst (subₙₜ₊ ξ 𝒟)
subₙₜ₊ ξ (snd 𝒟) = snd (subₙₜ₊ ξ 𝒟)
subₙₜ₊ ξ (enm 𝒟) = enm (subₙₘ₊ ξ 𝒟)
cutₙₘ₊ : ∀ {Γ A B} → Γ ⊢₊ A neutral → Γ , A ⊢₊ B normal
→ Γ ⊢₊ B normal
cutₙₘ₊ 𝒟 ℰ = subₙₘ₊ (idsₙₜ₊ , 𝒟) ℰ
pseudocutₙₘ₊ : ∀ {Γ A B} → Γ ⊢₊ A normal → Γ , A ⊢₊ B normal
→ Γ ⊢₊ B normal
pseudocutₙₘ₊ 𝒟 ℰ = ent (app (enm (lam ℰ)) 𝒟)
mutual
thm32ₙₘ : ∀ {Γ A} → Γ ⊢₊ A normal
→ Γ ⊢ A true
thm32ₙₘ (lam 𝒟) = lam (thm32ₙₘ 𝒟)
thm32ₙₘ (pair 𝒟 ℰ) = pair (thm32ₙₘ 𝒟) (thm32ₙₘ ℰ)
thm32ₙₘ unit = unit
thm32ₙₘ (abort 𝒟) = abort (thm32ₙₜ 𝒟)
thm32ₙₘ (inl 𝒟) = inl (thm32ₙₘ 𝒟)
thm32ₙₘ (inr 𝒟) = inr (thm32ₙₘ 𝒟)
thm32ₙₘ (case 𝒟 ℰ ℱ) = case (thm32ₙₜ 𝒟) (thm32ₙₘ ℰ) (thm32ₙₘ ℱ)
thm32ₙₘ (ent 𝒟) = thm32ₙₜ 𝒟
thm32ₙₜ : ∀ {Γ A} → Γ ⊢₊ A neutral
→ Γ ⊢ A true
thm32ₙₜ (var i) = var i
thm32ₙₜ (app 𝒟 ℰ) = app (thm32ₙₜ 𝒟) (thm32ₙₘ ℰ)
thm32ₙₜ (fst 𝒟) = fst (thm32ₙₜ 𝒟)
thm32ₙₜ (snd 𝒟) = snd (thm32ₙₜ 𝒟)
thm32ₙₜ (enm 𝒟) = thm32ₙₘ 𝒟
thm33ₙₘ : ∀ {Γ A} → Γ ⊢ A true
→ Γ ⊢₊ A normal
thm33ₙₘ (var i) = ent (var i)
thm33ₙₘ (lam 𝒟) = lam (thm33ₙₘ 𝒟)
thm33ₙₘ (app 𝒟 ℰ) = ent (app (enm (thm33ₙₘ 𝒟)) (thm33ₙₘ ℰ))
thm33ₙₘ (pair 𝒟 ℰ) = pair (thm33ₙₘ 𝒟) (thm33ₙₘ ℰ)
thm33ₙₘ (fst 𝒟) = ent (fst (enm (thm33ₙₘ 𝒟)))
thm33ₙₘ (snd 𝒟) = ent (snd (enm (thm33ₙₘ 𝒟)))
thm33ₙₘ unit = unit
thm33ₙₘ (abort 𝒟) = abort (enm (thm33ₙₘ 𝒟))
thm33ₙₘ (inl 𝒟) = inl (thm33ₙₘ 𝒟)
thm33ₙₘ (inr 𝒟) = inr (thm33ₙₘ 𝒟)
thm33ₙₘ (case 𝒟 ℰ ℱ) = case (enm (thm33ₙₘ 𝒟)) (thm33ₙₘ ℰ) (thm33ₙₘ ℱ)
thm33ₙₜ : ∀ {Γ A} → Γ ⊢ A true
→ Γ ⊢₊ A neutral
thm33ₙₜ (var i) = var i
thm33ₙₜ (lam 𝒟) = enm (lam (ent (thm33ₙₜ 𝒟)))
thm33ₙₜ (app 𝒟 ℰ) = app (thm33ₙₜ 𝒟) (ent (thm33ₙₜ ℰ))
thm33ₙₜ (pair 𝒟 ℰ) = enm (pair (ent (thm33ₙₜ 𝒟)) (ent (thm33ₙₜ ℰ)))
thm33ₙₜ (fst 𝒟) = fst (thm33ₙₜ 𝒟)
thm33ₙₜ (snd 𝒟) = snd (thm33ₙₜ 𝒟)
thm33ₙₜ unit = enm unit
thm33ₙₜ (abort 𝒟) = enm (abort (thm33ₙₜ 𝒟))
thm33ₙₜ (inl 𝒟) = enm (inl (ent (thm33ₙₜ 𝒟)))
thm33ₙₜ (inr 𝒟) = enm (inr (ent (thm33ₙₜ 𝒟)))
thm33ₙₜ (case 𝒟 ℰ ℱ) = enm (case (thm33ₙₜ 𝒟) (ent (thm33ₙₜ ℰ)) (ent (thm33ₙₜ ℱ)))