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Added Kovacs strong normalisation proof to extra

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Philip Wadler 2019-01-22 15:33:12 +00:00
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{-# OPTIONS --without-K #-}
open import Relation.Binary.PropositionalEquality
open import Data.Product
open import Data.Unit
open import Data.Empty
open import Function
-- some HoTT-inspired combinators
_&_ = cong
_⁻¹ = sym
_◾_ = trans
coe : {A B : Set} → A ≡ B → A → B
coe refl a = a
_⊗_ : ∀ {A B : Set}{f g : A → B}{a a'} → f ≡ g → a ≡ a' → f a ≡ g a'
refl ⊗ refl = refl
infix 6 _⁻¹
infixr 4 _◾_
infixl 9 _&_
infixl 8 _⊗_
-- Syntax
--------------------------------------------------------------------------------
infixr 4 _⇒_
infixr 4 _,_
data Ty : Set where
ι : Ty
_⇒_ : Ty → Ty → Ty
data Con : Set where
∙ : Con
_,_ : Con → Ty → Con
data _∈_ (A : Ty) : Con → Set where
vz : ∀ {Γ} → A ∈ (Γ , A)
vs : ∀ {B Γ} → A ∈ Γ → A ∈ (Γ , B)
data Tm Γ : Ty → Set where
var : ∀ {A} → A ∈ Γ → Tm Γ A
lam : ∀ {A B} → Tm (Γ , A) B → Tm Γ (A ⇒ B)
app : ∀ {A B} → Tm Γ (A ⇒ B) → Tm Γ A → Tm Γ B
-- Embedding
--------------------------------------------------------------------------------
-- Order-preserving embedding
data OPE : Con → Con → Set where
∙ : OPE ∙ ∙
drop : ∀ {A Γ Δ} → OPE Γ Δ → OPE (Γ , A) Δ
keep : ∀ {A Γ Δ} → OPE Γ Δ → OPE (Γ , A) (Δ , A)
-- OPE is a category
idₑ : ∀ {Γ} → OPE Γ Γ
idₑ {∙} = ∙
idₑ {Γ , A} = keep (idₑ {Γ})
wk : ∀ {A Γ} → OPE (Γ , A) Γ
wk = drop idₑ
_∘ₑ_ : ∀ {Γ Δ Σ} → OPE Δ Σ → OPE Γ Δ → OPE Γ Σ
σ ∘ₑ ∙ = σ
σ ∘ₑ drop δ = drop (σ ∘ₑ δ)
drop σ ∘ₑ keep δ = drop (σ ∘ₑ δ)
keep σ ∘ₑ keep δ = keep (σ ∘ₑ δ)
idlₑ : ∀ {Γ Δ}(σ : OPE Γ Δ) → idₑ ∘ₑ σ ≡ σ
idlₑ ∙ = refl
idlₑ (drop σ) = drop & idlₑ σ
idlₑ (keep σ) = keep & idlₑ σ
idrₑ : ∀ {Γ Δ}(σ : OPE Γ Δ) → σ ∘ₑ idₑ ≡ σ
idrₑ ∙ = refl
idrₑ (drop σ) = drop & idrₑ σ
idrₑ (keep σ) = keep & idrₑ σ
assₑ :
∀ {Γ Δ Σ Ξ}(σ : OPE Σ Ξ)(δ : OPE Δ Σ)(ν : OPE Γ Δ)
→ (σ ∘ₑ δ) ∘ₑ ν ≡ σ ∘ₑ (δ ∘ₑ ν)
assₑ σ δ ∙ = refl
assₑ σ δ (drop ν) = drop & assₑ σ δ ν
assₑ σ (drop δ) (keep ν) = drop & assₑ σ δ ν
assₑ (drop σ) (keep δ) (keep ν) = drop & assₑ σ δ ν
assₑ (keep σ) (keep δ) (keep ν) = keep & assₑ σ δ ν
∈ₑ : ∀ {A Γ Δ} → OPE Γ Δ → A ∈ Δ → A ∈ Γ
∈ₑ ∙ v = v
∈ₑ (drop σ) v = vs (∈ₑ σ v)
∈ₑ (keep σ) vz = vz
∈ₑ (keep σ) (vs v) = vs (∈ₑ σ v)
∈-idₑ : ∀ {A Γ}(v : A ∈ Γ) → ∈ₑ idₑ v ≡ v
∈-idₑ vz = refl
∈-idₑ (vs v) = vs & ∈-idₑ v
∈-∘ₑ : ∀ {A Γ Δ Σ}(σ : OPE Δ Σ)(δ : OPE Γ Δ)(v : A ∈ Σ) → ∈ₑ (σ ∘ₑ δ) v ≡ ∈ₑ δ (∈ₑ σ v)
∈-∘ₑ ∙ ∙ v = refl
∈-∘ₑ σ (drop δ) v = vs & ∈-∘ₑ σ δ v
∈-∘ₑ (drop σ) (keep δ) v = vs & ∈-∘ₑ σ δ v
∈-∘ₑ (keep σ) (keep δ) vz = refl
∈-∘ₑ (keep σ) (keep δ) (vs v) = vs & ∈-∘ₑ σ δ v
Tmₑ : ∀ {A Γ Δ} → OPE Γ Δ → Tm Δ A → Tm Γ A
Tmₑ σ (var v) = var (∈ₑ σ v)
Tmₑ σ (lam t) = lam (Tmₑ (keep σ) t)
Tmₑ σ (app f a) = app (Tmₑ σ f) (Tmₑ σ a)
Tm-idₑ : ∀ {A Γ}(t : Tm Γ A) → Tmₑ idₑ t ≡ t
Tm-idₑ (var v) = var & ∈-idₑ v
Tm-idₑ (lam t) = lam & Tm-idₑ t
Tm-idₑ (app f a) = app & Tm-idₑ f ⊗ Tm-idₑ a
Tm-∘ₑ : ∀ {A Γ Δ Σ}(σ : OPE Δ Σ)(δ : OPE Γ Δ)(t : Tm Σ A) → Tmₑ (σ ∘ₑ δ) t ≡ Tmₑ δ (Tmₑ σ t)
Tm-∘ₑ σ δ (var v) = var & ∈-∘ₑ σ δ v
Tm-∘ₑ σ δ (lam t) = lam & Tm-∘ₑ (keep σ) (keep δ) t
Tm-∘ₑ σ δ (app f a) = app & Tm-∘ₑ σ δ f ⊗ Tm-∘ₑ σ δ a
-- Theory of substitution & embedding
--------------------------------------------------------------------------------
infixr 6 _ₑ∘ₛ_ _ₛ∘ₑ_ _∘ₛ_
data Sub (Γ : Con) : Con → Set where
∙ : Sub Γ ∙
_,_ : ∀ {A : Ty}{Δ : Con} → Sub Γ Δ → Tm Γ A → Sub Γ (Δ , A)
_ₛ∘ₑ_ : ∀ {Γ Δ Σ} → Sub Δ Σ → OPE Γ Δ → Sub Γ Σ
∙ ₛ∘ₑ δ = ∙
(σ , t) ₛ∘ₑ δ = σ ₛ∘ₑ δ , Tmₑ δ t
_ₑ∘ₛ_ : ∀ {Γ Δ Σ} → OPE Δ Σ → Sub Γ Δ → Sub Γ Σ
∙ ₑ∘ₛ δ = δ
drop σ ₑ∘ₛ (δ , t) = σ ₑ∘ₛ δ
keep σ ₑ∘ₛ (δ , t) = σ ₑ∘ₛ δ , t
dropₛ : ∀ {A Γ Δ} → Sub Γ Δ → Sub (Γ , A) Δ
dropₛ σ = σ ₛ∘ₑ wk
keepₛ : ∀ {A Γ Δ} → Sub Γ Δ → Sub (Γ , A) (Δ , A)
keepₛ σ = dropₛ σ , var vz
⌜_⌝ᵒᵖᵉ : ∀ {Γ Δ} → OPE Γ Δ → Sub Γ Δ
⌜ ∙ ⌝ᵒᵖᵉ = ∙
⌜ drop σ ⌝ᵒᵖᵉ = dropₛ ⌜ σ ⌝ᵒᵖᵉ
⌜ keep σ ⌝ᵒᵖᵉ = keepₛ ⌜ σ ⌝ᵒᵖᵉ
∈ₛ : ∀ {A Γ Δ} → Sub Γ Δ → A ∈ Δ → Tm Γ A
∈ₛ (σ , t) vz = t
∈ₛ (σ , t)(vs v) = ∈ₛ σ v
Tmₛ : ∀ {A Γ Δ} → Sub Γ Δ → Tm Δ A → Tm Γ A
Tmₛ σ (var v) = ∈ₛ σ v
Tmₛ σ (lam t) = lam (Tmₛ (keepₛ σ) t)
Tmₛ σ (app f a) = app (Tmₛ σ f) (Tmₛ σ a)
idₛ : ∀ {Γ} → Sub Γ Γ
idₛ {∙} = ∙
idₛ {Γ , A} = (idₛ {Γ} ₛ∘ₑ drop idₑ) , var vz
_∘ₛ_ : ∀ {Γ Δ Σ} → Sub Δ Σ → Sub Γ Δ → Sub Γ Σ
∙ ∘ₛ δ = ∙
(σ , t) ∘ₛ δ = σ ∘ₛ δ , Tmₛ δ t
assₛₑₑ :
∀ {Γ Δ Σ Ξ}(σ : Sub Σ Ξ)(δ : OPE Δ Σ)(ν : OPE Γ Δ)
→ (σ ₛ∘ₑ δ) ₛ∘ₑ ν ≡ σ ₛ∘ₑ (δ ∘ₑ ν)
assₛₑₑ ∙ δ ν = refl
assₛₑₑ (σ , t) δ ν = _,_ & assₛₑₑ σ δ ν ⊗ (Tm-∘ₑ δ ν t ⁻¹)
assₑₛₑ :
∀ {Γ Δ Σ Ξ}(σ : OPE Σ Ξ)(δ : Sub Δ Σ)(ν : OPE Γ Δ)
→ (σ ₑ∘ₛ δ) ₛ∘ₑ ν ≡ σ ₑ∘ₛ (δ ₛ∘ₑ ν)
assₑₛₑ ∙ δ ν = refl
assₑₛₑ (drop σ) (δ , t) ν = assₑₛₑ σ δ ν
assₑₛₑ (keep σ) (δ , t) ν = (_, Tmₑ ν t) & assₑₛₑ σ δ ν
idlₑₛ : ∀ {Γ Δ}(σ : Sub Γ Δ) → idₑ ₑ∘ₛ σ ≡ σ
idlₑₛ ∙ = refl
idlₑₛ (σ , t) = (_, t) & idlₑₛ σ
idlₛₑ : ∀ {Γ Δ}(σ : OPE Γ Δ) → idₛ ₛ∘ₑ σ ≡ ⌜ σ ⌝ᵒᵖᵉ
idlₛₑ ∙ = refl
idlₛₑ (drop σ) =
((idₛ ₛ∘ₑ_) ∘ drop) & idrₑ σ ⁻¹
◾ assₛₑₑ idₛ σ wk ⁻¹
◾ dropₛ & idlₛₑ σ
idlₛₑ (keep σ) =
(_, var vz) &
(assₛₑₑ idₛ wk (keep σ)
◾ ((idₛ ₛ∘ₑ_) ∘ drop) & (idlₑ σ ◾ idrₑ σ ⁻¹)
◾ assₛₑₑ idₛ σ wk ⁻¹
◾ (_ₛ∘ₑ wk) & idlₛₑ σ )
idrₑₛ : ∀ {Γ Δ}(σ : OPE Γ Δ) → σ ₑ∘ₛ idₛ ≡ ⌜ σ ⌝ᵒᵖᵉ
idrₑₛ ∙ = refl
idrₑₛ (drop σ) = assₑₛₑ σ idₛ wk ⁻¹ ◾ dropₛ & idrₑₛ σ
idrₑₛ (keep σ) = (_, var vz) & (assₑₛₑ σ idₛ wk ⁻¹ ◾ (_ₛ∘ₑ wk) & idrₑₛ σ)
∈-ₑ∘ₛ : ∀ {A Γ Δ Σ}(σ : OPE Δ Σ)(δ : Sub Γ Δ)(v : A ∈ Σ) → ∈ₛ (σ ₑ∘ₛ δ) v ≡ ∈ₛ δ (∈ₑ σ v)
∈-ₑ∘ₛ ∙ δ v = refl
∈-ₑ∘ₛ (drop σ) (δ , t) v = ∈-ₑ∘ₛ σ δ v
∈-ₑ∘ₛ (keep σ) (δ , t) vz = refl
∈-ₑ∘ₛ (keep σ) (δ , t) (vs v) = ∈-ₑ∘ₛ σ δ v
Tm-ₑ∘ₛ : ∀ {A Γ Δ Σ}(σ : OPE Δ Σ)(δ : Sub Γ Δ)(t : Tm Σ A) → Tmₛ (σ ₑ∘ₛ δ) t ≡ Tmₛ δ (Tmₑ σ t)
Tm-ₑ∘ₛ σ δ (var v) = ∈-ₑ∘ₛ σ δ v
Tm-ₑ∘ₛ σ δ (lam t) =
lam & ((λ x → Tmₛ (x , var vz) t) & assₑₛₑ σ δ wk ◾ Tm-ₑ∘ₛ (keep σ) (keepₛ δ) t)
Tm-ₑ∘ₛ σ δ (app f a) = app & Tm-ₑ∘ₛ σ δ f ⊗ Tm-ₑ∘ₛ σ δ a
∈-ₛ∘ₑ : ∀ {A Γ Δ Σ}(σ : Sub Δ Σ)(δ : OPE Γ Δ)(v : A ∈ Σ) → ∈ₛ (σ ₛ∘ₑ δ) v ≡ Tmₑ δ (∈ₛ σ v)
∈-ₛ∘ₑ (σ , _) δ vz = refl
∈-ₛ∘ₑ (σ , _) δ (vs v) = ∈-ₛ∘ₑ σ δ v
Tm-ₛ∘ₑ : ∀ {A Γ Δ Σ}(σ : Sub Δ Σ)(δ : OPE Γ Δ)(t : Tm Σ A) → Tmₛ (σ ₛ∘ₑ δ) t ≡ Tmₑ δ (Tmₛ σ t)
Tm-ₛ∘ₑ σ δ (var v) = ∈-ₛ∘ₑ σ δ v
Tm-ₛ∘ₑ σ δ (lam t) =
lam &
((λ x → Tmₛ (x , var vz) t) &
(assₛₑₑ σ δ wk
◾ (σ ₛ∘ₑ_) & (drop & (idrₑ δ ◾ idlₑ δ ⁻¹))
◾ assₛₑₑ σ wk (keep δ) ⁻¹)
◾ Tm-ₛ∘ₑ (keepₛ σ) (keep δ) t)
Tm-ₛ∘ₑ σ δ (app f a) = app & Tm-ₛ∘ₑ σ δ f ⊗ Tm-ₛ∘ₑ σ δ a
assₛₑₛ :
∀ {Γ Δ Σ Ξ}(σ : Sub Σ Ξ)(δ : OPE Δ Σ)(ν : Sub Γ Δ)
→ (σ ₛ∘ₑ δ) ∘ₛ ν ≡ σ ∘ₛ (δ ₑ∘ₛ ν)
assₛₑₛ ∙ δ ν = refl
assₛₑₛ (σ , t) δ ν = _,_ & assₛₑₛ σ δ ν ⊗ (Tm-ₑ∘ₛ δ ν t ⁻¹)
assₛₛₑ :
∀ {Γ Δ Σ Ξ}(σ : Sub Σ Ξ)(δ : Sub Δ Σ)(ν : OPE Γ Δ)
→ (σ ∘ₛ δ) ₛ∘ₑ ν ≡ σ ∘ₛ (δ ₛ∘ₑ ν)
assₛₛₑ ∙ δ ν = refl
assₛₛₑ (σ , t) δ ν = _,_ & assₛₛₑ σ δ ν ⊗ (Tm-ₛ∘ₑ δ ν t ⁻¹)
∈-idₛ : ∀ {A Γ}(v : A ∈ Γ) → ∈ₛ idₛ v ≡ var v
∈-idₛ vz = refl
∈-idₛ (vs v) = ∈-ₛ∘ₑ idₛ wk v ◾ Tmₑ wk & ∈-idₛ v ◾ (var ∘ vs) & ∈-idₑ v
∈-∘ₛ : ∀ {A Γ Δ Σ}(σ : Sub Δ Σ)(δ : Sub Γ Δ)(v : A ∈ Σ) → ∈ₛ (σ ∘ₛ δ) v ≡ Tmₛ δ (∈ₛ σ v)
∈-∘ₛ (σ , _) δ vz = refl
∈-∘ₛ (σ , _) δ (vs v) = ∈-∘ₛ σ δ v
Tm-idₛ : ∀ {A Γ}(t : Tm Γ A) → Tmₛ idₛ t ≡ t
Tm-idₛ (var v) = ∈-idₛ v
Tm-idₛ (lam t) = lam & Tm-idₛ t
Tm-idₛ (app f a) = app & Tm-idₛ f ⊗ Tm-idₛ a
Tm-∘ₛ : ∀ {A Γ Δ Σ}(σ : Sub Δ Σ)(δ : Sub Γ Δ)(t : Tm Σ A) → Tmₛ (σ ∘ₛ δ) t ≡ Tmₛ δ (Tmₛ σ t)
Tm-∘ₛ σ δ (var v) = ∈-∘ₛ σ δ v
Tm-∘ₛ σ δ (lam t) =
lam &
((λ x → Tmₛ (x , var vz) t) &
(assₛₛₑ σ δ wk
◾ (σ ∘ₛ_) & (idlₑₛ (dropₛ δ) ⁻¹) ◾ assₛₑₛ σ wk (keepₛ δ) ⁻¹)
◾ Tm-∘ₛ (keepₛ σ) (keepₛ δ) t)
Tm-∘ₛ σ δ (app f a) = app & Tm-∘ₛ σ δ f ⊗ Tm-∘ₛ σ δ a
idrₛ : ∀ {Γ Δ}(σ : Sub Γ Δ) → σ ∘ₛ idₛ ≡ σ
idrₛ ∙ = refl
idrₛ (σ , t) = _,_ & idrₛ σ ⊗ Tm-idₛ t
idlₛ : ∀ {Γ Δ}(σ : Sub Γ Δ) → idₛ ∘ₛ σ ≡ σ
idlₛ ∙ = refl
idlₛ (σ , t) = (_, t) & (assₛₑₛ idₛ wk (σ , t) ◾ (idₛ ∘ₛ_) & idlₑₛ σ ◾ idlₛ σ)
-- Reduction
--------------------------------------------------------------------------------
data _~>_ {Γ} : ∀ {A} → Tm Γ A → Tm Γ A → Set where
β : ∀ {A B}(t : Tm (Γ , A) B) t' → app (lam t) t' ~> Tmₛ (idₛ , t') t
lam : ∀ {A B}{t t' : Tm (Γ , A) B} → t ~> t' → lam t ~> lam t'
app₁ : ∀ {A B}{f}{f' : Tm Γ (A ⇒ B)}{a} → f ~> f' → app f a ~> app f' a
app₂ : ∀ {A B}{f : Tm Γ (A ⇒ B)} {a a'} → a ~> a' → app f a ~> app f a'
infix 3 _~>_
~>ₛ : ∀ {Γ Δ A}{t t' : Tm Γ A}(σ : Sub Δ Γ) → t ~> t' → Tmₛ σ t ~> Tmₛ σ t'
~>ₛ σ (β t t') =
coe ((app (lam (Tmₛ (keepₛ σ) t)) (Tmₛ σ t') ~>_) &
(Tm-∘ₛ (keepₛ σ) (idₛ , Tmₛ σ t') t ⁻¹
◾ (λ x → Tmₛ (x , Tmₛ σ t') t) &
(assₛₑₛ σ wk (idₛ , Tmₛ σ t')
◾ ((σ ∘ₛ_) & idlₑₛ idₛ ◾ idrₛ σ) ◾ idlₛ σ ⁻¹)
◾ Tm-∘ₛ (idₛ , t') σ t))
(β (Tmₛ (keepₛ σ) t) (Tmₛ σ t'))
~>ₛ σ (lam step) = lam (~>ₛ (keepₛ σ) step)
~>ₛ σ (app₁ step) = app₁ (~>ₛ σ step)
~>ₛ σ (app₂ step) = app₂ (~>ₛ σ step)
~>ₑ : ∀ {Γ Δ A}{t t' : Tm Γ A}(σ : OPE Δ Γ) → t ~> t' → Tmₑ σ t ~> Tmₑ σ t'
~>ₑ σ (β t t') =
coe ((app (lam (Tmₑ (keep σ) t)) (Tmₑ σ t') ~>_)
& (Tm-ₑ∘ₛ (keep σ) (idₛ , Tmₑ σ t') t ⁻¹
◾ (λ x → Tmₛ (x , Tmₑ σ t') t) & (idrₑₛ σ ◾ idlₛₑ σ ⁻¹)
◾ Tm-ₛ∘ₑ (idₛ , t') σ t))
(β (Tmₑ (keep σ) t) (Tmₑ σ t'))
~>ₑ σ (lam step) = lam (~>ₑ (keep σ) step)
~>ₑ σ (app₁ step) = app₁ (~>ₑ σ step)
~>ₑ σ (app₂ step) = app₂ (~>ₑ σ step)
Tmₑ~> :
∀ {Γ Δ A}{t : Tm Γ A}{σ : OPE Δ Γ}{t'}
→ Tmₑ σ t ~> t' → ∃ λ t'' → (t ~> t'') × (Tmₑ σ t'' ≡ t')
Tmₑ~> {t = var x} ()
Tmₑ~> {t = lam t} (lam step) with Tmₑ~> step
... | t'' , (p , refl) = lam t'' , lam p , refl
Tmₑ~> {t = app (var v) a} (app₁ ())
Tmₑ~> {t = app (var v) a} (app₂ step) with Tmₑ~> step
... | t'' , (p , refl) = app (var v) t'' , app₂ p , refl
Tmₑ~> {t = app (lam f) a} {σ} (β _ _) =
Tmₛ (idₛ , a) f , β _ _ ,
Tm-ₛ∘ₑ (idₛ , a) σ f ⁻¹
◾ (λ x → Tmₛ (x , Tmₑ σ a) f) & (idlₛₑ σ ◾ idrₑₛ σ ⁻¹)
◾ Tm-ₑ∘ₛ (keep σ) (idₛ , Tmₑ σ a) f
Tmₑ~> {t = app (lam f) a} (app₁ (lam step)) with Tmₑ~> step
... | t'' , (p , refl) = app (lam t'') a , app₁ (lam p) , refl
Tmₑ~> {t = app (lam f) a} (app₂ step) with Tmₑ~> step
... | t'' , (p , refl) = app (lam f) t'' , app₂ p , refl
Tmₑ~> {t = app (app f a) a'} (app₁ step) with Tmₑ~> step
... | t'' , (p , refl) = app t'' a' , app₁ p , refl
Tmₑ~> {t = app (app f a) a''} (app₂ step) with Tmₑ~> step
... | t'' , (p , refl) = app (app f a) t'' , app₂ p , refl
-- Strong normalization/neutrality definition
--------------------------------------------------------------------------------
data SN {Γ A} (t : Tm Γ A) : Set where
sn : (∀ {t'} → t ~> t' → SN t') → SN t
SNₑ→ : ∀ {Γ Δ A}{t : Tm Γ A}(σ : OPE Δ Γ) → SN t → SN (Tmₑ σ t)
SNₑ→ σ (sn s) = sn λ {t'} step →
let (t'' , (p , q)) = Tmₑ~> step in coe (SN & q) (SNₑ→ σ (s p))
SNₑ← : ∀ {Γ Δ A}{t : Tm Γ A}(σ : OPE Δ Γ) → SN (Tmₑ σ t) → SN t
SNₑ← σ (sn s) = sn λ step → SNₑ← σ (s (~>ₑ σ step))
SN-app₁ : ∀ {Γ A B}{f : Tm Γ (A ⇒ B)}{a} → SN (app f a) → SN f
SN-app₁ (sn s) = sn λ f~>f' → SN-app₁ (s (app₁ f~>f'))
neu : ∀ {Γ A} → Tm Γ A → Set
neu (lam _) = ⊥
neu _ = ⊤
neuₑ : ∀ {Γ Δ A}(σ : OPE Δ Γ)(t : Tm Γ A) → neu t → neu (Tmₑ σ t)
neuₑ σ (lam t) nt = nt
neuₑ σ (var v) nt = tt
neuₑ σ (app f a) nt = tt
-- The actual proof, by Kripke logical predicate
--------------------------------------------------------------------------------
Tmᴾ : ∀ {Γ A} → Tm Γ A → Set
Tmᴾ {Γ}{ι} t = SN t
Tmᴾ {Γ}{A ⇒ B} t = ∀ {Δ}(σ : OPE Δ Γ){a} → Tmᴾ a → Tmᴾ (app (Tmₑ σ t) a)
data Subᴾ {Γ} : ∀ {Δ} → Sub Γ Δ → Set where
∙ : Subᴾ ∙
_,_ : ∀ {A Δ}{σ : Sub Γ Δ}{t : Tm Γ A}(σᴾ : Subᴾ σ)(tᴾ : Tmᴾ t) → Subᴾ (σ , t)
Tmᴾₑ : ∀ {Γ Δ A}{t : Tm Γ A}(σ : OPE Δ Γ) → Tmᴾ t → Tmᴾ (Tmₑ σ t)
Tmᴾₑ {A = ι} σ tᴾ = SNₑ→ σ tᴾ
Tmᴾₑ {A = A ⇒ B}{t} σ tᴾ δ aᴾ rewrite Tm-∘ₑ σ δ t ⁻¹ = tᴾ (σ ∘ₑ δ) aᴾ
Subᴾₑ : ∀ {Γ Δ Σ}{σ : Sub Δ Σ}(δ : OPE Γ Δ) → Subᴾ σ → Subᴾ (σ ₛ∘ₑ δ)
Subᴾₑ σ ∙ = ∙
Subᴾₑ σ (δ , tᴾ) = Subᴾₑ σ δ , Tmᴾₑ σ tᴾ
~>ᴾ : ∀ {Γ A}{t t' : Tm Γ A} → t ~> t' → Tmᴾ t → Tmᴾ t'
~>ᴾ {A = ι} t~>t' (sn tˢⁿ) = tˢⁿ t~>t'
~>ᴾ {A = A ⇒ B} t~>t' tᴾ = λ σ aᴾ → ~>ᴾ (app₁ (~>ₑ σ t~>t')) (tᴾ σ aᴾ)
mutual
-- quote
qᴾ : ∀ {Γ A}{t : Tm Γ A} → Tmᴾ t → SN t
qᴾ {A = ι} tᴾ = tᴾ
qᴾ {A = A ⇒ B} tᴾ = SNₑ← wk $ SN-app₁ (qᴾ $ tᴾ wk (uᴾ (var vz) (λ ())))
-- unquote
uᴾ : ∀ {Γ A}(t : Tm Γ A){nt : neu t} → (∀ {t'} → t ~> t' → Tmᴾ t') → Tmᴾ t
uᴾ {Γ} {A = ι} t f = sn f
uᴾ {Γ} {A ⇒ B} t {nt} f {Δ} σ {a} aᴾ =
uᴾ (app (Tmₑ σ t) a) (go (Tmₑ σ t) (neuₑ σ t nt) f' a aᴾ (qᴾ aᴾ))
where
f' : ∀ {t'} → Tmₑ σ t ~> t' → Tmᴾ t'
f' step δ aᴾ with Tmₑ~> step
... | t'' , step' , refl rewrite Tm-∘ₑ σ δ t'' ⁻¹ = f step' (σ ∘ₑ δ) aᴾ
go :
∀ {Γ A B}(t : Tm Γ (A ⇒ B)) → neu t → (∀ {t'} → t ~> t' → Tmᴾ t')
→ ∀ a → Tmᴾ a → SN a → ∀ {t'} → app t a ~> t' → Tmᴾ t'
go _ () _ _ _ _ (β _ _)
go t nt f a aᴾ sna (app₁ {f' = f'} step) =
coe ((λ x → Tmᴾ (app x a)) & Tm-idₑ f') (f step idₑ aᴾ)
go t nt f a aᴾ (sn aˢⁿ) (app₂ {a' = a'} step) =
uᴾ (app t a') (go t nt f a' (~>ᴾ step aᴾ) (aˢⁿ step))
fundThm-∈ : ∀ {Γ A}(v : A ∈ Γ) → ∀ {Δ}{σ : Sub Δ Γ} → Subᴾ σ → Tmᴾ (∈ₛ σ v)
fundThm-∈ vz (σᴾ , tᴾ) = tᴾ
fundThm-∈ (vs v) (σᴾ , tᴾ) = fundThm-∈ v σᴾ
fundThm-lam :
∀ {Γ A B}
(t : Tm (Γ , A) B)
→ SN t
→ (∀ {a} → Tmᴾ a → Tmᴾ (Tmₛ (idₛ , a) t))
→ ∀ a → SN a → Tmᴾ a → Tmᴾ (app (lam t) a)
fundThm-lam {Γ} t (sn tˢⁿ) hyp a (sn aˢⁿ) aᴾ = uᴾ (app (lam t) a)
λ {(β _ _) → hyp aᴾ;
(app₁ (lam {t' = t'} t~>t')) →
fundThm-lam t' (tˢⁿ t~>t') (λ aᴾ → ~>ᴾ (~>ₛ _ t~>t') (hyp aᴾ)) a (sn aˢⁿ) aᴾ;
(app₂ a~>a') →
fundThm-lam t (sn tˢⁿ) hyp _ (aˢⁿ a~>a') (~>ᴾ a~>a' aᴾ)}
fundThm : ∀ {Γ A}(t : Tm Γ A) → ∀ {Δ}{σ : Sub Δ Γ} → Subᴾ σ → Tmᴾ (Tmₛ σ t)
fundThm (var v) σᴾ = fundThm-∈ v σᴾ
fundThm (lam {A} t) {σ = σ} σᴾ δ {a} aᴾ
rewrite Tm-ₛ∘ₑ (keepₛ σ) (keep δ) t ⁻¹ | assₛₑₑ σ (wk {A}) (keep δ) | idlₑ δ
= fundThm-lam
(Tmₛ (σ ₛ∘ₑ drop δ , var vz) t)
(qᴾ (fundThm t (Subᴾₑ (drop δ) σᴾ , uᴾ (var vz) (λ ()))))
(λ aᴾ → coe (Tmᴾ & sub-sub-lem) (fundThm t (Subᴾₑ δ σᴾ , aᴾ)))
a (qᴾ aᴾ) aᴾ
where
sub-sub-lem : ∀ {a} → Tmₛ (σ ₛ∘ₑ δ , a) t ≡ Tmₛ (idₛ , a) (Tmₛ (σ ₛ∘ₑ drop δ , var vz) t)
sub-sub-lem {a} =
(λ x → Tmₛ (x , a) t) &
(idrₛ (σ ₛ∘ₑ δ) ⁻¹ ◾ assₛₑₛ σ δ idₛ ◾ assₛₑₛ σ (drop δ) (idₛ , a) ⁻¹)
◾ Tm-∘ₛ (σ ₛ∘ₑ drop δ , var vz) (idₛ , a) t
fundThm (app f a) {σ = σ} σᴾ
rewrite Tm-idₑ (Tmₛ σ f) ⁻¹
= fundThm f σᴾ idₑ (fundThm a σᴾ)
idₛᴾ : ∀ {Γ} → Subᴾ (idₛ {Γ})
idₛᴾ {∙} = ∙
idₛᴾ {Γ , A} = Subᴾₑ wk idₛᴾ , uᴾ (var vz) (λ ())
strongNorm : ∀ {Γ A}(t : Tm Γ A) → SN t
strongNorm t = qᴾ (coe (Tmᴾ & Tm-idₛ t) (fundThm t idₛᴾ))