{-# OPTIONS --without-K #-}

{-
Below is an example of proving *everything* about the substitution calculus of a
simple TT with Bool.

The more challenging solution:
  - If substitutions are defined as lists of terms, then if you manage to
    prove that substitutions form a category, almost certainly you're done,
    since you can only prove this is you have all relevant lemmas.

The lazy solution:
  - Copy/paste everything from below, then add/remove cases as necessary,
    depending on the definition of the syntax.
-}

-- Equational reasoning
--------------------------------------------------------------------------------

open import Relation.Binary.PropositionalEquality
open import Function

_◾_ : ∀ {α}{A : Set α}{x y z : A} → x ≡ y → y ≡ z → x ≡ z
refl ◾ refl = refl
infixr 4 _◾_

_⁻¹ : ∀{α}{A : Set α}{x y : A} → x ≡ y → y ≡ x
refl ⁻¹ = refl
infix 6 _⁻¹

coe : ∀{α}{A B : Set α} → A ≡ B → A → B
coe refl a = a

_&_ :
  ∀{α β}{A : Set α}{B : Set β}(f : A → B){a₀ a₁ : A}(a₂ : a₀ ≡ a₁)
  → f a₀ ≡ f a₁
f & refl = refl
infixl 9 _&_

_⊗_ :
  ∀ {α β}{A : Set α}{B : Set β}{f g : A → B}(p : f ≡ g){a a' : A}(q : a ≡ a')
  → f a ≡ g a'
refl ⊗ refl = refl
infixl 8 _⊗_

-- Syntax
--------------------------------------------------------------------------------

infixr 4 _⇒_
infixr 5 _,_

data Ty : Set where
  Bool : 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
  true  : Tm Γ Bool
  false : Tm Γ Bool
  bcase : ∀ {A} → Tm Γ Bool → Tm Γ A → Tm Γ A → Tm Γ A

-- Order-preserving embeddings
--------------------------------------------------------------------------------

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)

-- Category operations
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)

-- Category laws
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

-- (A ∈ _) : PSh OPE ("PSh C" means contravariant functor from C to Set)
∈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

-- (Tm _ A) : PSh OPE
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)
Tmₑ σ true          = true
Tmₑ σ false         = false
Tmₑ σ (bcase b t f) = bcase (Tmₑ σ b) (Tmₑ σ t) (Tmₑ σ f)

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-idₑ true          = refl
Tm-idₑ false         = refl
Tm-idₑ (bcase b t f) = bcase & Tm-idₑ b ⊗ Tm-idₑ t ⊗ Tm-idₑ f

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
Tm-∘ₑ σ δ true          = refl
Tm-∘ₑ σ δ false         = refl
Tm-∘ₑ σ δ (bcase b t f) = bcase & Tm-∘ₑ σ δ b ⊗ Tm-∘ₑ σ δ t ⊗ Tm-∘ₑ σ δ f

-- Substitution
--------------------------------------------------------------------------------

infixr 6 _ₑ∘ₛ_ _ₛ∘ₑ_ _∘ₛ_

data Sub (Γ : Con) : Con → Set where
  ∙   : Sub Γ ∙
  _,_ : ∀ {A Δ} → Sub Γ Δ → Tm Γ A → Sub Γ (Δ , A)

-- left and right compositions
_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)
Tmₛ σ true          = true
Tmₛ σ false         = false
Tmₛ σ (bcase b t f) = bcase (Tmₛ σ b) (Tmₛ σ t) (Tmₛ σ f)

idₛ : ∀ {Γ} → Sub Γ Γ
idₛ {∙}     = ∙
idₛ {Γ , A} = keepₛ idₛ

_∘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

-- Various versions of associativity and identity
-- with substitution/embedding differing on sides of operations.
-- Both for category laws and Tm/∈ functor laws.
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
Tm-ₑ∘ₛ σ δ true          = refl
Tm-ₑ∘ₛ σ δ false         = refl
Tm-ₑ∘ₛ σ δ (bcase b t f) = bcase & Tm-ₑ∘ₛ σ δ b ⊗ Tm-ₑ∘ₛ σ δ t ⊗ Tm-ₑ∘ₛ σ δ f

∈-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
Tm-ₛ∘ₑ σ δ true          = refl
Tm-ₛ∘ₑ σ δ false         = refl
Tm-ₛ∘ₑ σ δ (bcase b t f) = bcase & Tm-ₛ∘ₑ σ δ b ⊗ Tm-ₛ∘ₑ σ δ t ⊗ Tm-ₛ∘ₑ σ δ f

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-idₛ true          = refl
Tm-idₛ false         = refl
Tm-idₛ (bcase b t f) = bcase & Tm-idₛ b ⊗ Tm-idₛ t ⊗ Tm-idₛ f

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
Tm-∘ₛ σ δ true          = refl
Tm-∘ₛ σ δ false         = refl
Tm-∘ₛ σ δ (bcase b t f) = bcase & Tm-∘ₛ σ δ b ⊗ Tm-∘ₛ σ δ t ⊗ Tm-∘ₛ σ δ f

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)

assₛ :
  ∀ {Γ Δ Σ Ξ}(σ : Sub Σ Ξ)(δ : Sub Δ Σ)(ν : Sub Γ Δ)
  → (σ ∘ₛ δ) ∘ₛ ν ≡ σ ∘ₛ (δ ∘ₛ ν)
assₛ ∙       δ ν = refl
assₛ (σ , t) δ ν = _,_ & assₛ σ δ ν ⊗ (Tm-∘ₛ δ ν t ⁻¹)