FFOLInitial2.lagda

git s\begin{code}
{-# OPTIONS --prop --rewriting #-}

open import PropUtil

module FFOLInitial2 where

  open import FFOL
  open import Agda.Primitive
  open import ListUtil

  data Con : Set₁
  data TmVar : Con → Set₁
  data Tm  : Con → Set₁
  data For : Con → Set₁
  data Con where
    ◇ : Con
    _▹ₜ : Con → Con
    _▹ₚ_ : (Γ : Con) → For Γ → Con

  variable
    Γ Δ Ξ : Con
    
  -- A term variable is a de-bruijn variable, TmVar n ≈ ⟦0,n-1⟧
  data TmVar where
    tvzero : TmVar (Γ ▹ₜ)
    tvnext : TmVar Γ → TmVar (Γ ▹ₜ)

  -- For now, we only have term variables (no function symbol)
  data Tm where
    var : TmVar Γ → Tm Γ
    
  -- Now we can define formulæ
  data For where
    R : Tm Γ → Tm Γ → For Γ
    _⇒_ : For Γ → For Γ → For Γ
    ∀∀  : For (Γ ▹ₜ) → For Γ

  -------------------------------------------------------------------


  -------------------------------------------------------------------
  
  data Sub : Con → Con → Set₁
  data Ren : Con → Con → Set₁
  id : Sub Γ Γ
  idᵣ : Ren Γ Γ
  _∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
  _∘ᵣ_ : {Γ Δ Ξ : Con} → Ren Δ Ξ → Ren Γ Δ → Ren Γ Ξ
  _[_]t : Tm Γ → Sub Δ Γ → Tm Δ
  _[_]f : For Γ → Sub Δ Γ → For Δ
  _[_]tvᵣ : TmVar Γ → Ren Δ Γ → TmVar Δ
  _[_]tᵣ : Tm Γ → Ren Δ Γ → Tm Δ
  _[_]fᵣ : For Γ → Ren Δ Γ → For Δ
  []f-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ}{β : Sub Δ Γ}{A : For Γ} → A [ β ∘ α ]f ≡ (A [ β ]f) [ α ]f
  []t-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ}{β : Sub Δ Γ}{t : Tm Γ} → t [ β ∘ α ]t ≡ (t [ β ]t) [ α ]t
  []fᵣ-∘ᵣ : {Γ Δ Ξ : Con} → {α : Ren Ξ Δ}{β : Ren Δ Γ}{A : For Γ} → A [ β ∘ᵣ α ]fᵣ ≡ (A [ β ]fᵣ) [ α ]fᵣ
  []tᵣ-∘ᵣ : {Γ Δ Ξ : Con} → {α : Ren Ξ Δ}{β : Ren Δ Γ}{t : Tm Γ} → t [ β ∘ᵣ α ]tᵣ ≡ (t [ β ]tᵣ) [ α ]tᵣ
  data PfVar : (Γ : Con) → For Γ → Prop₁
  data Pf : (Γ : Con) → For Γ → Prop₁
  πₜ¹ : Sub Δ (Γ ▹ₜ) → Sub Δ Γ
  πₜ² : Sub Δ (Γ ▹ₜ) → Tm Δ
  πₚ¹ : {Γ Δ : Con} {A : For Γ} → Sub Δ (Γ ▹ₚ A) → Sub Δ Γ
  πₚ² : {Γ Δ : Con} {A : For Γ} → (σ : Sub Δ (Γ ▹ₚ A)) → Pf Δ (A [ πₚ¹ σ ]f)
  _[_]p : {Γ Δ : Con} → {A : For Γ} → Pf Γ A → (σ : Sub Δ Γ) → Pf Δ (A [ σ ]f)
  _[_]pvᵣ : {Γ Δ : Con} → {A : For Γ} → PfVar Γ A → (σ : Ren Δ Γ) → PfVar Δ (A [ σ ]fᵣ)
 
  data Sub where
    ε : Sub Γ ◇
    _,ₜ_ : Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
    _,ₚ_ : {A : For Γ} → (σ : Sub Δ Γ) → Pf Δ (A [ σ ]f) → Sub Δ (Γ ▹ₚ A)

  πₜ¹ (σ ,ₜ t) = σ
  πₜ² (σ ,ₜ t) = t
  πₚ¹ (σ ,ₚ pf) = σ
  πₚ² (σ ,ₚ pf) = pf


  data Ren where
    εᵣ : Ren Γ ◇
    _,ₜᵣ_ : Ren Δ Γ → TmVar Δ → Ren Δ (Γ ▹ₜ)
    _,ₚᵣ_ : {A : For Γ} → (σ : Ren Δ Γ) → PfVar Δ (A [ σ ]fᵣ) → Ren Δ (Γ ▹ₚ A)

  πₜ¹ᵣ : Ren Δ (Γ ▹ₜ) → Ren Δ Γ
  πₜ²ᵣ : Ren Δ (Γ ▹ₜ) → TmVar Δ
  πₚ¹ᵣ : {Γ Δ : Con} {A : For Γ} → Ren Δ (Γ ▹ₚ A) → Ren Δ Γ
  πₚ²ᵣ : {Γ Δ : Con} {A : For Γ} → (σ : Ren Δ (Γ ▹ₚ A)) → PfVar Δ (A [ πₚ¹ᵣ σ ]fᵣ)
  πₜ¹ᵣ (σ ,ₜᵣ t) = σ
  πₜ²ᵣ (σ ,ₜᵣ t) = t
  πₚ¹ᵣ (σ ,ₚᵣ pf) = σ
  πₚ²ᵣ (σ ,ₚᵣ pf) = pf

  liftᵣₜ : Ren Δ Γ → Ren (Δ ▹ₜ) (Γ ▹ₜ)
  liftᵣₜ εᵣ = εᵣ ,ₜᵣ tvzero
  liftᵣₜ (σ ,ₜᵣ t) = (liftᵣₜ σ) ,ₜᵣ tvzero
  liftᵣₜ (σ ,ₚᵣ p) = {!!}
  idᵣ {◇} = εᵣ
  idᵣ {Γ ▹ₜ} = liftᵣₜ (idᵣ {Γ})
  idᵣ {Γ ▹ₚ A} = {!!}
  
  εᵣ ∘ᵣ β = εᵣ
  (α ,ₜᵣ t) ∘ᵣ β = (α ∘ᵣ β) ,ₜᵣ (t [ β ]tvᵣ)
  (α ,ₚᵣ pf) ∘ᵣ β = (α ∘ᵣ β) ,ₚᵣ substP (PfVar _) (≡sym []fᵣ-∘ᵣ) (pf [ β ]pvᵣ)

  tvzero [ _ ,ₜᵣ tv ]tvᵣ = tv
  tvnext tv [ σ ,ₜᵣ _ ]tvᵣ = tv [ σ ]tvᵣ
  var tv [ σ ]tᵣ = var (tv [ σ ]tvᵣ)
  (R t u) [ σ ]fᵣ = R (t [ σ ]tᵣ) (u [ σ ]tᵣ)
  (A ⇒ B) [ σ ]fᵣ = (A [ σ ]fᵣ) ⇒ (B [ σ ]fᵣ)
  (∀∀ A) [ σ ]fᵣ = ∀∀ (A [ (σ ∘ᵣ πₜ¹ᵣ idᵣ) ,ₜᵣ (πₜ²ᵣ idᵣ) ]fᵣ)


  {-
  -- We now show how we can extend renamings
  rightRen :{A : For Δₜ} → Ren Γₚ Δₚ → Ren Γₚ (Δₚ ▹p⁰ A)
  rightRen zeroRen = zeroRen
  rightRen (leftRen x h) = leftRen (pvnext x) (rightRen h)
  bothRen : {A : For Γₜ} → Ren Γₚ Δₚ → Ren (Γₚ ▹p⁰ A) (Δₚ ▹p⁰ A)
  bothRen zeroRen = leftRen pvzero zeroRen
  bothRen (leftRen x h) = leftRen pvzero (leftRen (pvnext x) (rightRen h))
  reflRen : Ren Γₚ Γₚ
  reflRen {Γₚ = ◇p} = zeroRen
  reflRen {Γₚ = Γₚ ▹p⁰ x} = bothRen reflRen
  -}
  
  id = {!!}
  ε ∘ β = ε
  (α ,ₜ t) ∘ β = (α ∘ β) ,ₜ (t [ β ]t)
  (α ,ₚ pf) ∘ β = (α ∘ β) ,ₚ substP (Pf _) (≡sym []f-∘) (pf [ β ]p)
  
  var tvzero [ σ ,ₜ t ]t = t
  var (tvnext tv) [ σ ,ₜ t ]t = var tv [ σ ]t
  (R t u) [ σ ]f = R (t [ σ ]t) (u [ σ ]t)
  (A ⇒ B) [ σ ]f = (A [ σ ]f) ⇒ (B [ σ ]f)
  (∀∀ A) [ σ ]f = ∀∀ (A [ (σ ∘ πₜ¹ id) ,ₜ πₜ² id ]f)

  []t-∘ {β = β ,ₜ t} {t = var tvzero} = {!!}
  []t-∘ {t = var (tvnext tv)} = {!!}
  []f-∘ {A = R t u} = {!cong₂ R!}
  []f-∘ {A = A ⇒ B} = {!!}
  []f-∘ {A = ∀∀ A} = {!!}
  


  data PfVar where
    pvzero : {A : For Γ} → PfVar (Γ ▹ₚ A) (A [ πₚ¹ id ]f)
    pvnext : {A B : For Γ} → PfVar Γ A → PfVar (Γ ▹ₚ B) (A [ πₚ¹ id ]f)
  data Pf where
    var : {A : For Γ} → PfVar Γ A → Pf Γ A
    app : {A B : For Γ} → Pf Γ (A ⇒ B) → Pf Γ A → Pf Γ B
    lam : {A B : For Γ} → Pf (Γ ▹ₚ A) (B [ πₚ¹ id ]f) → Pf Γ (A ⇒ B)
    p∀∀e : {A : For (Γ ▹ₜ)} → {t : Tm Γ} → Pf Γ (∀∀ A) → Pf Γ (A [ id ,ₜ t ]f)
    p∀∀i : {A : For (Γ ▹ₜ)} → Pf (Γ ▹ₜ) A → Pf Γ (∀∀ A)

  var pvzero [ σ ,ₚ pf ]p = {!pf!}
  var (pvnext pv) [ σ ]p = {!!}
  app pf pf' [ σ ]p = {!!}
  lam pf [ σ ]p = {!!}
  p∀∀e pf [ σ ]p = {!!}
  p∀∀i pf [ σ ]p = {!!}
  
  {-
  {-- TERM CONTEXTS - TERMS - FORMULAE - TERM SUBSTITUTIONS --}

  -- Term contexts are isomorphic to Nat
  data Cont : Set₁ where
    ◇t : Cont
    _▹t⁰ : Cont → Cont
    
  variable
    Γₜ Δₜ Ξₜ : Cont





  -- Then we define term substitutions
    
  -- We write down the access functions from the algebra, in restricted versions
  -- And their equalities (the fact that there are reciprocical)
  πₜ²∘,ₜ : {σₜ : Subt Δₜ Γₜ} → {t : Tm Δₜ} → πₜ² (σₜ ,ₜ t) ≡ t
  πₜ²∘,ₜ = refl
  πₜ¹∘,ₜ : {σₜ : Subt Δₜ Γₜ} → {t : Tm Δₜ} → πₜ¹ (σₜ ,ₜ t) ≡ σₜ
  πₜ¹∘,ₜ = refl
  ,ₜ∘πₜ : {σₜ : Subt Δₜ (Γₜ ▹t⁰)} → (πₜ¹ σₜ) ,ₜ (πₜ² σₜ) ≡ σₜ
  ,ₜ∘πₜ {σₜ = σₜ ,ₜ t} = refl

    
  -- We now define the action of term substitutions on terms
  
  -- We define weakenings of the term-context for terms
  -- «A term of n variables can be seen as a term of n+1 variables»
  wkₜt : Tm Γₜ → Tm (Γₜ ▹t⁰)
  wkₜt (var tv) = var (tvnext tv)
  
  -- From a substition into n variables, we get a substitution into n+1 variables which don't use the last one
  wkₜσₜ : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) Γₜ
  wkₜσₜ εₜ = εₜ
  wkₜσₜ (σ ,ₜ t) = (wkₜσₜ σ) ,ₜ (wkₜt t)
  
  -- From a substitution into n variables, we construct a substitution from n+1 variables to n+1 variables which maps it to itself
  -- i.e. 0 -> 0 and for all i ->(old) σ(i) we get i+1 -> σ(i)+1
  lfₜσₜ : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) (Γₜ ▹t⁰)
  lfₜσₜ σ = (wkₜσₜ σ) ,ₜ (var tvzero)

  -- We show how wkₜt and interacts with [_]t
  wkₜ[]t : {α : Subt Δₜ Γₜ} → {t : Tm Γₜ} →  wkₜt (t [ α ]t) ≡ (wkₜt t [ lfₜσₜ α ]t)
  wkₜ[]t {α = α ,ₜ t} {var tvzero} = refl
  wkₜ[]t {α = α ,ₜ t} {var (tvnext tv)} = wkₜ[]t {t = var tv}
  
  -- We can now subst on formulæ




  -- We now can define identity and composition of term substitutions       
  idₜ : Subt Γₜ Γₜ
  idₜ {◇t} = εₜ
  idₜ {Γₜ ▹t⁰} = lfₜσₜ (idₜ {Γₜ})
  _∘ₜ_ : Subt Δₜ Γₜ → Subt Ξₜ Δₜ → Subt Ξₜ Γₜ
  εₜ ∘ₜ β = εₜ
  (α ,ₜ x) ∘ₜ β = (α ∘ₜ β) ,ₜ (x [ β ]t)

  -- We now have to show all their equalities (idₜ and ∘ₜ respect []t, []f, wkₜ, lfₜ, categorical rules

  -- Substitution for terms
  []t-id : {t : Tm Γₜ} → t [ idₜ {Γₜ} ]t ≡ t
  []t-id {Γₜ ▹t⁰} {var tvzero} = refl
  []t-id {Γₜ ▹t⁰} {var (tvnext tv)} = substP (λ t → t ≡ var (tvnext tv)) (wkₜ[]t {t = var tv}) (substP (λ t → wkₜt t ≡ var (tvnext tv)) (≡sym ([]t-id {t = var tv})) refl)
  []t-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → {t : Tm Γₜ} → t [ β ∘ₜ α ]t ≡ (t [ β ]t) [ α ]t
  []t-∘ {α = α} {β = β ,ₜ t} {t = var tvzero} = refl
  []t-∘ {α = α} {β = β ,ₜ t} {t = var (tvnext tv)} = []t-∘ {t = var tv}
  
  -- Weakenings and liftings of substitutions
  wkₜσₜ-∘ₜl : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → wkₜσₜ (β ∘ₜ α) ≡ (wkₜσₜ β ∘ₜ lfₜσₜ α)
  wkₜσₜ-∘ₜl {β = εₜ} = refl
  wkₜσₜ-∘ₜl {β = β ,ₜ t} = cong₂ _,ₜ_ wkₜσₜ-∘ₜl (wkₜ[]t {t = t})
  wkₜσₜ-∘ₜr : {α : Subt Γₜ Δₜ} → {β : Subt Ξₜ Γₜ} → α ∘ₜ (wkₜσₜ β) ≡ wkₜσₜ (α ∘ₜ β)
  wkₜσₜ-∘ₜr {α = εₜ} = refl
  wkₜσₜ-∘ₜr {α = α ,ₜ var tv} = cong₂ _,ₜ_ (wkₜσₜ-∘ₜr {α = α}) (≡sym (wkₜ[]t {t = var tv}))
  lfₜσₜ-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → lfₜσₜ (β ∘ₜ α) ≡ (lfₜσₜ β) ∘ₜ (lfₜσₜ α)
  lfₜσₜ-∘ {α = α} {β = εₜ} = refl
  lfₜσₜ-∘ {α = α} {β = β ,ₜ t} = cong₂ _,ₜ_ (cong₂ _,ₜ_ wkₜσₜ-∘ₜl (wkₜ[]t {t = t})) refl

  -- Cancelling a weakening with a ,ₜ
  wkₜ[,]t : {t : Tm Γₜ}{u : Tm Δₜ}{β : Subt Δₜ Γₜ} → (wkₜt t) [ β ,ₜ u ]t ≡ t [ β ]t
  wkₜ[,]t {t = var tvzero} = refl
  wkₜ[,]t {t = var (tvnext tv)} = refl
  wkₜ∘ₜ,ₜ : {α : Subt Γₜ Δₜ}{β : Subt Ξₜ Γₜ}{t : Tm Ξₜ} → (wkₜσₜ α) ∘ₜ (β ,ₜ t) ≡ (α ∘ₜ β)
  wkₜ∘ₜ,ₜ {α = εₜ} = refl
  wkₜ∘ₜ,ₜ {α = α ,ₜ t} {β = β} = cong₂ _,ₜ_ (wkₜ∘ₜ,ₜ {α = α}) (wkₜ[,]t {t = t} {β = β})
  
  -- Categorical rules are respected by idₜ and ∘ₜ
  idlₜ : {α : Subt Δₜ Γₜ} → idₜ ∘ₜ α ≡ α
  idlₜ {α = εₜ} = refl
  idlₜ {α = α ,ₜ x} = cong₂ _,ₜ_ (≡tran wkₜ∘ₜ,ₜ idlₜ) refl
  idrₜ : {α : Subt Δₜ Γₜ} → α ∘ₜ idₜ ≡ α
  idrₜ {α = εₜ} = refl
  idrₜ {α = α ,ₜ x} = cong₂ _,ₜ_ idrₜ []t-id
  ∘ₜ-ass : {Γₜ Δₜ Ξₜ Ψₜ : Cont}{α : Subt Γₜ Δₜ}{β : Subt Δₜ Ξₜ}{γ : Subt Ξₜ Ψₜ} → (γ ∘ₜ β) ∘ₜ α ≡ γ ∘ₜ (β ∘ₜ α)
  ∘ₜ-ass {α = α} {β} {εₜ} = refl
  ∘ₜ-ass {α = α} {β} {γ ,ₜ x} = cong₂ _,ₜ_ ∘ₜ-ass (≡sym ([]t-∘ {t = x}))
  
  -- Unicity of the terminal morphism
  εₜ-u : {σₜ : Subt Γₜ ◇t} → σₜ ≡ εₜ
  εₜ-u {σₜ = εₜ} = refl

  -- Substitution for formulæ
  []f-id : {F : For Γₜ} → F [ idₜ {Γₜ} ]f ≡ F
  []f-id {F = R t u} = cong₂ R []t-id []t-id
  []f-id {F = F ⇒ G} = cong₂ _⇒_ []f-id []f-id
  []f-id {F = ∀∀ F} = cong ∀∀ []f-id
  []f-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → {F : For Γₜ} → F [ β ∘ₜ α ]f ≡ (F [ β ]f) [ α ]f
  []f-∘ {α = α} {β = β} {F = R t u} = cong₂ R ([]t-∘ {α = α} {β = β} {t = t}) ([]t-∘ {α = α} {β = β} {t = u})
  []f-∘ {F = F ⇒ G} = cong₂ _⇒_ []f-∘ []f-∘
  []f-∘ {F = ∀∀ F} = cong ∀∀ (≡tran (cong (λ σ → F [ σ ]f) lfₜσₜ-∘) []f-∘)

  -- Substitution for formulæ constructors
  -- we omit []f-R and []f-⇒ as they are directly refl
  []f-∀∀ : {A : For (Γₜ ▹t⁰)} → {σₜ : Subt Δₜ Γₜ} → (∀∀ A) [ σₜ ]f ≡ (∀∀ (A [ (σₜ ∘ₜ πₜ¹ idₜ) ,ₜ πₜ² idₜ ]f))
  []f-∀∀ {A = A} = cong ∀∀ (cong (_[_]f A) (cong₂ _,ₜ_ (≡tran (cong wkₜσₜ (≡sym idrₜ)) (≡sym wkₜσₜ-∘ₜr)) refl))








  -- We can now define proof contexts, which are indexed by a term context
  -- i.e. we know which terms a proof context can use
  data Conp : Cont → Set₁ where
    ◇p : Conp Γₜ
    _▹p⁰_ : Conp Γₜ → For Γₜ → Conp Γₜ
  
  variable
    Γₚ Γₚ' : Conp Γₜ
    Δₚ Δₚ' : Conp Δₜ
    Ξₚ Ξₚ' : Conp Ξₜ

  -- The actions of Subt's is extended to contexts
  _[_]c : Conp Γₜ → Subt Δₜ Γₜ → Conp Δₜ
  ◇p [ σₜ ]c = ◇p
  (Γₚ ▹p⁰ A) [ σₜ ]c = (Γₚ [ σₜ ]c) ▹p⁰ (A [ σₜ ]f)
  -- This Conp is indeed a functor
  []c-id : Γₚ [ idₜ ]c ≡ Γₚ
  []c-id {Γₚ = ◇p} = refl
  []c-id {Γₚ = Γₚ ▹p⁰ x} = cong₂ _▹p⁰_ []c-id []f-id
  []c-∘ : {α : Subt Δₜ Ξₜ} {β : Subt Γₜ Δₜ} {Ξₚ : Conp Ξₜ} → Ξₚ [ α ∘ₜ β ]c ≡ (Ξₚ [ α ]c) [ β ]c 
  []c-∘ {α = α} {β = β} {◇p} = refl
  []c-∘ {α = α} {β = β} {Ξₚ ▹p⁰ A} = cong₂ _▹p⁰_ []c-∘ []f-∘


  -- We can also add a term that will not be used in the formulæ already present
  -- (that's why we use wkₜσₜ)
  _▹tp : Conp Γₜ → Conp (Γₜ ▹t⁰)
  Γ ▹tp = Γ [ wkₜσₜ idₜ ]c
  -- We show how it interacts with ,ₜ and lfₜσₜ
  ▹tp,ₜ : {σₜ : Subt Γₜ Δₜ}{t : Tm Γₜ} → (Γₚ ▹tp) [ σₜ ,ₜ t ]c ≡  Γₚ [ σₜ ]c
  ▹tp,ₜ {Γₚ = Γₚ} = ≡tran (≡sym []c-∘) (cong (λ ξ → Γₚ [ ξ ]c) (≡tran wkₜ∘ₜ,ₜ idlₜ))
  ▹tp-lfₜ : {σ : Subt Δₜ Γₜ} → ((Δₚ ▹tp) [ lfₜσₜ σ ]c) ≡ ((Δₚ [ σ ]c) ▹tp)
  ▹tp-lfₜ {Δₚ = Δₚ} = ≡tran² (≡sym []c-∘) (cong (λ ξ → Δₚ [ ξ ]c) (≡tran² (≡sym wkₜσₜ-∘ₜl) (cong wkₜσₜ (≡tran idlₜ (≡sym idrₜ))) (≡sym wkₜσₜ-∘ₜr))) []c-∘



  -- With those contexts, we have everything to define proofs


  -- The action on Cont's morphisms of Pf functor
  _[_]pvₜ : {A : For Δₜ}→ PfVar Δₜ Δₚ A → (σ : Subt Γₜ Δₜ)→ PfVar Γₜ (Δₚ [ σ ]c) (A [ σ ]f)
  pvzero [ σ ]pvₜ = pvzero
  pvnext pv [ σ ]pvₜ = pvnext (pv [ σ ]pvₜ)
  _[_]pₜ : {A : For Δₜ} → Pf Δₜ Δₚ A → (σ : Subt Γₜ Δₜ) → Pf Γₜ (Δₚ [ σ ]c) (A [ σ ]f)
  var pv [ σ ]pₜ = var (pv [ σ ]pvₜ)
  app pf pf' [ σ ]pₜ = app (pf [ σ ]pₜ) (pf' [ σ ]pₜ)
  lam pf [ σ ]pₜ = lam (pf [ σ ]pₜ)
  _[_]pₜ {Δₚ = Δₚ} {Γₜ = Γₜ} (p∀∀e {A = A} {t = t} pf) σ =
    substP (λ F → Pf Γₜ (Δₚ [ σ ]c) F) (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f)
    (cong₂ _,ₜ_ (≡tran² wkₜ∘ₜ,ₜ idrₜ (≡sym idlₜ)) refl)) ([]f-∘))
    (p∀∀e {t = t [ σ ]t} (pf [ σ ]pₜ))
  _[_]pₜ {Γₜ = Γₜ} (p∀∀i pf) σ
    = p∀∀i (substP (λ Ξₚ → Pf (Γₜ ▹t⁰) (Ξₚ) _) ▹tp-lfₜ (pf [ lfₜσₜ σ ]pₜ))
  



  

  -- We now can create Renamings, a subcategory from (Conp,Subp) that
  -- A renaming from a context Γₚ to a context Δₚ means when they are seen
  -- as lists, that every element of Γₚ is an element of Δₚ
  -- In other words, we can prove Γₚ from Δₚ using only proof variables (var)

  -- We can extend renamings with term variables
  PfVar▹tp : {A : For Δₜ} → PfVar Δₜ Δₚ A → PfVar (Δₜ ▹t⁰) (Δₚ ▹tp) (A [ wkₜσₜ idₜ ]f)
  PfVar▹tp pvzero = pvzero
  PfVar▹tp (pvnext x) = pvnext (PfVar▹tp x)
  Ren▹tp : Ren Γₚ Δₚ → Ren (Γₚ ▹tp) (Δₚ ▹tp)
  Ren▹tp zeroRen = zeroRen
  Ren▹tp (leftRen x s) = leftRen (PfVar▹tp x) (Ren▹tp s)

  -- Renamings can be used to (strongly) weaken proofs
  wkᵣpv : {A : For Δₜ} → Ren Δₚ' Δₚ → PfVar Δₜ Δₚ' A → PfVar Δₜ Δₚ A
  wkᵣpv (leftRen x x₁) pvzero = x
  wkᵣpv (leftRen x x₁) (pvnext s) = wkᵣpv x₁ s
  wkᵣp : {A : For Δₜ} → Ren Δₚ Δₚ' → Pf Δₜ Δₚ A → Pf Δₜ Δₚ' A
  wkᵣp s (var pv) = var (wkᵣpv s pv)
  wkᵣp s (app pf pf₁) = app (wkᵣp s pf) (wkᵣp s pf₁)
  wkᵣp s (lam {A = A} pf) = lam (wkᵣp (bothRen s) pf)
  wkᵣp s (p∀∀e pf) = p∀∀e (wkᵣp s pf)
  wkᵣp s (p∀∀i pf) = p∀∀i (wkᵣp (Ren▹tp s) pf)


  -- But we need something stronger than just renamings
  -- introducing: Proof substitutions
  -- They are basicly a list of proofs for the formulæ contained in
  -- the goal context.
  -- It is not defined between all contexts, only those with the same term context
  data Subp : {Δₜ : Cont} → Conp Δₜ → Conp Δₜ → Prop₁ where
    εₚ : Subp Δₚ ◇p

    
  -- We write down the access functions from the algebra, in restricted versions

  -- The action of Cont's morphisms on Subp
  _[_]σₚ : Subp {Δₜ} Δₚ Δₚ' → (σ : Subt Γₜ Δₜ) → Subp {Γₜ} (Δₚ [ σ ]c) (Δₚ' [ σ ]c)
  εₚ [ σₜ ]σₚ = εₚ
  (σₚ ,ₚ pf) [ σₜ ]σₚ = (σₚ [ σₜ ]σₚ) ,ₚ (pf [ σₜ ]pₜ)
  
  -- They are indeed stronger than renamings
  Ren→Sub : Ren Δₚ Δₚ' → Subp {Δₜ} Δₚ' Δₚ
  Ren→Sub zeroRen = εₚ
  Ren→Sub (leftRen x s) = Ren→Sub s ,ₚ var x


  -- From a substition into n variables, we get a substitution into n+1 variables which don't use the last one
  wkₚσₚ : {Δₜ : Cont} {Δₚ Γₚ : Conp Δₜ}{A : For Δₜ} → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} (Δₚ ▹p⁰ A) Γₚ
  wkₚσₚ εₚ = εₚ
  wkₚσₚ (σₚ ,ₚ pf) = (wkₚσₚ σₚ) ,ₚ wkᵣp (rightRen reflRen) pf
  
  -- From a substitution into n variables, we construct a substitution from n+1 variables to n+1 variables which maps it to itself
  -- i.e. 0 -> 0 and for all i ->(old) σ(i) we get i+1 -> σ(i)+1
  lfₚσₚ : {Δₜ : Cont}{Δₚ Γₚ : Conp Δₜ}{A : For Δₜ} → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} (Δₚ ▹p⁰ A) (Γₚ ▹p⁰ A)
  lfₚσₚ σ = (wkₚσₚ σ) ,ₚ (var pvzero)

  
  





  

  wkₜσₚ : Subp {Δₜ} Δₚ' Δₚ → Subp {Δₜ ▹t⁰} (Δₚ' ▹tp) (Δₚ ▹tp)
  wkₜσₚ εₚ = εₚ
  wkₜσₚ {Δₜ = Δₜ} (_,ₚ_ {A = A} σₚ pf) = (wkₜσₚ σₚ) ,ₚ substP (λ Ξₚ → Pf (Δₜ ▹t⁰) Ξₚ (A [ wkₜσₜ idₜ ]f)) refl (_[_]pₜ {Γₜ = Δₜ ▹t⁰} pf (wkₜσₜ idₜ))

  _[_]p : {A : For Δₜ} → Pf Δₜ Δₚ A → (σ : Subp {Δₜ} Δₚ' Δₚ) → Pf Δₜ Δₚ' A
  var pvzero [ σ ,ₚ pf ]p = pf
  var (pvnext pv) [ σ ,ₚ pf ]p = var pv [ σ ]p
  app pf pf₁ [ σ ]p = app (pf [ σ ]p) (pf₁ [ σ ]p)
  lam pf [ σ ]p = lam (pf [ wkₚσₚ σ ,ₚ var pvzero ]p) 
  p∀∀e pf [ σ ]p = p∀∀e (pf [ σ ]p)
  p∀∀i pf [ σ ]p = p∀∀i (pf [ wkₜσₚ σ ]p)







  -- We can now define identity and composition on proof substitutions
  idₚ : Subp {Δₜ} Δₚ Δₚ
  idₚ {Δₚ = ◇p} = εₚ
  idₚ {Δₚ = Δₚ ▹p⁰ x} = lfₚσₚ (idₚ {Δₚ = Δₚ})









  -- We can now merge the two notions of contexts, substitutions, and everything
  record Con : Set₁ where
    constructor con
    field
      t : Cont
      p : Conp t
      
  variable
    Γ Δ Ξ : Con
    
  record Sub (Γ : Con) (Δ : Con) : Set₁ where
    constructor sub
    field
      t : Subt (Con.t Γ) (Con.t Δ)
      p : Subp {Con.t Γ} (Con.p Γ) ((Con.p Δ) [ t ]c)

  -- We need this to apply term-substitution theorems to global substitutions
  sub= : {Γ Δ : Con}{σₜ σₜ' : Subt (Con.t Γ) (Con.t Δ)} →
    σₜ ≡ σₜ' →
    {σₚ : Subp {Con.t Γ} (Con.p Γ) ((Con.p Δ) [ σₜ ]c)}
    {σₚ' : Subp {Con.t Γ} (Con.p Γ) ((Con.p Δ) [ σₜ' ]c)} → 
    sub σₜ σₚ ≡ sub σₜ' σₚ'
  sub= refl = refl 

  -- (Con,Sub) is a category with an initial object
  id : Sub Γ Γ
  id {Γ} = sub idₜ (substP (Subp _) (≡sym []c-id) idₚ)
  _∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
  sub αₜ αₚ ∘ sub βₜ βₚ = sub (αₜ ∘ₜ βₜ) (substP (Subp _) (≡sym []c-∘) (αₚ [ βₜ ]σₚ) ∘ₚ βₚ)
    

  -- We have our two context extension operators
  _▹t : Con → Con
  Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp)
  _▹p_ : (Γ : Con) → For (Con.t Γ) → Con
  Γ ▹p A = con (Con.t Γ) (Con.p Γ ▹p⁰ A)

  -- We define the access function from the algebra, but defined for fully-featured substitutions
  -- For term substitutions
  πₜ¹* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Sub Δ Γ
  πₜ¹* (sub (σₜ ,ₜ t) σₚ) = sub σₜ (substP (Subp _) ▹tp,ₜ σₚ)
  πₜ²* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Tm (Con.t Δ)
  πₜ²* (sub (σₜ ,ₜ t) σₚ) = t
  _,ₜ*_ : {Γ Δ : Con} → Sub Δ Γ → Tm (Con.t Δ) → Sub Δ (Γ ▹t)
  (sub σₜ σₚ) ,ₜ* t = sub (σₜ ,ₜ t) (substP (Subp _) (≡sym ▹tp,ₜ) σₚ)
  -- And the equations
  πₜ²∘,ₜ* : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm (Con.t Δ)} → πₜ²* (σ ,ₜ* t) ≡ t
  πₜ²∘,ₜ* = refl
  πₜ¹∘,ₜ* : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm (Con.t Δ)} → πₜ¹* (σ ,ₜ* t) ≡ σ
  πₜ¹∘,ₜ* {Γ}{Δ}{σ}{t} = sub= refl
  ,ₜ∘πₜ* : {Γ Δ : Con} → {σ : Sub Δ (Γ ▹t)} → (πₜ¹* σ) ,ₜ* (πₜ²* σ) ≡ σ
  ,ₜ∘πₜ* {Γ} {Δ} {sub (σₜ ,ₜ t) σₚ} = sub= refl
  ,ₜ∘* : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{t : Tm (Con.t Γ)} → (σ ,ₜ* t) ∘ δ ≡ (σ ∘ δ) ,ₜ* (t [ Sub.t δ ]t)
  ,ₜ∘* {Γ} {Δ} {Ξ} {sub σₜ σₚ} {sub δₜ δₚ} {t} = sub= refl


  -- And for proof substitutions
  πₚ¹* : {Γ Δ : Con} {A : For (Con.t Γ)} → Sub Δ (Γ ▹p A) → Sub Δ Γ
  πₚ¹* (sub σₜ σaₚ) = sub σₜ (πₚ¹ σaₚ)
  πₚ²* : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ (Γ ▹p F)) → Pf (Con.t Δ) (Con.p Δ) (F [ Sub.t (πₚ¹* σ) ]f)
  πₚ²* (sub σₜ (σₚ ,ₚ pf)) = pf
  _,ₚ*_ : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ Γ) → Pf (Con.t Δ) (Con.p Δ) (F [ Sub.t σ ]f) → Sub Δ (Γ ▹p F)
  sub σₜ σₚ ,ₚ* pf = sub σₜ (σₚ ,ₚ pf)
  -- And the equations
  ,ₚ∘πₚ : {Γ Δ : Con} → {F : For (Con.t Γ)} → {σ : Sub Δ (Γ ▹p F)} → (πₚ¹* σ) ,ₚ* (πₚ²* σ) ≡ σ
  ,ₚ∘πₚ {σ = sub σₜ (σₚ ,ₚ p)} = refl
  ,ₚ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{F : For (Con.t Ξ)}{prf : Pf (Con.t Γ) (Con.p Γ) (F [ Sub.t σ ]f)}
      → (σ ,ₚ* prf) ∘ δ ≡ (σ ∘ δ) ,ₚ* (substP (λ F → Pf (Con.t Δ) (Con.p Δ) F) (≡sym []f-∘) ((prf [ Sub.t δ ]pₜ) [ Sub.p δ ]p))
  ,ₚ∘ {Γ}{Δ}{Ξ}{σ = sub σₜ σₚ} {sub δₜ δₚ} {F = A} {prf} = sub= refl
  
  -- and FINALLY, we compile everything into an implementation of the FFOL record
    
  ffol : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero}
  ffol = record
    { Con = Con
    ; Sub = Sub
    ; _∘_ = _∘_
    ; ∘-ass = sub= ∘ₜ-ass
    ; id = id
    ; idl = sub= idlₜ
    ; idr = sub= idrₜ
    ; ◇ = con ◇t ◇p
    ; ε = sub εₜ εₚ
    ; ε-u = sub= εₜ-u
    ; Tm = λ Γ → Tm (Con.t Γ)
    ; _[_]t = λ t σ → t [ Sub.t σ ]t
    ; []t-id = []t-id
    ; []t-∘ = λ {Γ}{Δ}{Ξ}{α}{β}{t} → []t-∘ {α = Sub.t α} {β = Sub.t β} {t = t}
    ; _▹ₜ = _▹t
    ; πₜ¹ = πₜ¹*
    ; πₜ² = πₜ²*
    ; _,ₜ_ = _,ₜ*_
    ; πₜ²∘,ₜ = refl
    ; πₜ¹∘,ₜ = λ {Γ}{Δ}{σ}{t} → πₜ¹∘,ₜ* {Γ}{Δ}{σ}{t}
    ; ,ₜ∘πₜ = ,ₜ∘πₜ*
    ; ,ₜ∘ = λ {Γ}{Δ}{Ξ}{σ}{δ}{t} → ,ₜ∘* {Γ}{Δ}{Ξ}{σ}{δ}{t}
    ; For = λ Γ → For (Con.t Γ)
    ; _[_]f = λ A σ → A [ Sub.t σ ]f
    ; []f-id = []f-id
    ; []f-∘ = []f-∘
    ; R = R
    ; R[] = refl
    ; _⊢_ = λ Γ A → Pf (Con.t Γ) (Con.p Γ) A
    ; _[_]p = λ pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p
    ; _▹ₚ_ = _▹p_
    ; πₚ¹ = πₚ¹*
    ; πₚ² = πₚ²*
    ; _,ₚ_ = _,ₚ*_
    ; ,ₚ∘πₚ = ,ₚ∘πₚ
    ; πₚ¹∘,ₚ = refl
    ; ,ₚ∘ = λ {Γ}{Δ}{Ξ}{σ}{δ}{F}{prf} → ,ₚ∘ {Γ}{Δ}{Ξ}{σ}{δ}{F}{prf}
    ; _⇒_ = _⇒_
    ; []f-⇒ = refl
    ; ∀∀ = ∀∀
    ; []f-∀∀ = []f-∀∀
    ; lam = λ {Γ}{F}{G} pf → substP (λ H → Pf (Con.t Γ) (Con.p Γ) (F ⇒ H)) []f-id (lam pf)
    ; app = app
    ; ∀i = p∀∀i
    ; ∀e = λ {Γ} {F} pf {t} → p∀∀e pf
    }


  -- We define normal and neutral forms
  data Ne : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Prop₁
  data Nf : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Prop₁
  data Ne where
    var : {A : For Γₜ} → PfVar Γₜ Γₚ A → Ne Γₜ Γₚ A
    app : {A B : For Γₜ} → Ne Γₜ Γₚ (A ⇒ B) → Nf Γₜ Γₚ A → Ne Γₜ Γₚ B
    p∀∀e : {A : For (Γₜ ▹t⁰)} → {t : Tm Γₜ} → Ne Γₜ Γₚ (∀∀ A) → Ne Γₜ Γₚ (A [ idₜ ,ₜ t ]f)
  data Nf where
    R : {t u : Tm Γₜ} → Ne Γₜ Γₚ (R t u) → Nf Γₜ Γₚ (R t u)
    lam : {A B : For Γₜ} → Nf Γₜ (Γₚ ▹p⁰ A) B → Nf Γₜ Γₚ (A ⇒ B)
    p∀∀i : {A : For (Γₜ ▹t⁰)} → Nf (Γₜ ▹t⁰) (Γₚ ▹tp) A → Nf Γₜ Γₚ (∀∀ A)


  Pf* : (Γₜ : Cont) → Conp Γₜ → Conp Γₜ → Prop₁
  Pf* Γₜ Γₚ ◇p = ⊤
  Pf* Γₜ Γₚ (Γₚ' ▹p⁰ A) = (Pf* Γₜ Γₚ Γₚ') ∧ (Pf Γₜ Γₚ A)

  Sub→Pf* : {Γₜ : Cont} {Γₚ Γₚ' : Conp Γₜ} →  Subp {Γₜ} Γₚ Γₚ' → Pf* Γₜ Γₚ Γₚ'
  Sub→Pf* εₚ = tt
  Sub→Pf* (σₚ ,ₚ pf) = ⟨ (Sub→Pf* σₚ) , pf ⟩
  Pf*-id : {Γₜ : Cont} {Γₚ : Conp Γₜ} → Pf* Γₜ Γₚ Γₚ
  Pf*-id = Sub→Pf* idₚ

  Pf*▹p : {Γₜ : Cont}{Γₚ Γₚ' : Conp Γₜ}{A : For Γₜ} → Pf* Γₜ Γₚ Γₚ' → Pf* Γₜ (Γₚ ▹p⁰ A) Γₚ'
  Pf*▹p {Γₚ' = ◇p} s = tt
  Pf*▹p {Γₚ' = Γₚ' ▹p⁰ x} s = ⟨ (Pf*▹p (proj₁ s)) , (wkᵣp (rightRen reflRen) (proj₂ s)) ⟩
  Pf*▹tp : {Γₜ : Cont}{Γₚ Γₚ' : Conp Γₜ} → Pf* Γₜ Γₚ Γₚ' → Pf* (Γₜ ▹t⁰) (Γₚ ▹tp) (Γₚ' ▹tp)
  Pf*▹tp {Γₚ' = ◇p} s = tt
  Pf*▹tp {Γₚ' = Γₚ' ▹p⁰ A} s = ⟨ Pf*▹tp (proj₁ s) , (proj₂ s) [ wkₜσₜ idₜ ]pₜ ⟩

  Pf*Pf : {Γₜ : Cont} {Γₚ Γₚ' : Conp Γₜ} {A : For Γₜ} → Pf* Γₜ Γₚ Γₚ' → Pf Γₜ Γₚ' A → Pf Γₜ Γₚ A
  Pf*Pf s (var pvzero) = proj₂ s
  Pf*Pf s (var (pvnext pv)) = Pf*Pf (proj₁ s) (var pv)
  Pf*Pf s (app p p') = app (Pf*Pf s p) (Pf*Pf s p')
  Pf*Pf s (lam p) = lam (Pf*Pf (⟨ (Pf*▹p s) , (var pvzero) ⟩) p)
  Pf*Pf s (p∀∀e p) = p∀∀e (Pf*Pf s p)
  Pf*Pf s (p∀∀i p) = p∀∀i (Pf*Pf (Pf*▹tp s) p)

  Pf*-∘ : {Γₜ : Cont} {Γₚ Δₚ Ξₚ : Conp Γₜ} → Pf* Γₜ Δₚ Ξₚ → Pf* Γₜ Γₚ Δₚ → Pf* Γₜ Γₚ Ξₚ
  Pf*-∘ {Ξₚ = ◇p} α β = tt
  Pf*-∘ {Ξₚ = Ξₚ ▹p⁰ A} α β = ⟨ Pf*-∘ (proj₁ α) β , Pf*Pf β (proj₂ α) ⟩
  -}

  {-
  module InitialMorphism (M : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero}) where
    {-# TERMINATING #-}

    mCont : Cont → (FFOL.Con M)
    mCont ◇t = FFOL.◇ M
    mCont (Γₜ ▹t⁰) = FFOL._▹ₜ M (mCont Γₜ)
    mTmT : {Γₜ : Cont} → Tm Γₜ → (FFOL.Tm M (mCont Γₜ))
    -- Zero is (πₜ² id)
    mTmT {Γₜ ▹t⁰} (var tvzero) = FFOL.πₜ² M (FFOL.id M)
    -- N+1 is wk[tm N]
    mTmT {Γₜ ▹t⁰} (var (tvnext tv)) = (FFOL._[_]t M (mTmT (var tv)) (FFOL.πₜ¹ M (FFOL.id M)))

    mForT : {Γₜ : Cont} → (For Γₜ) → (FFOL.For M (mCont Γₜ))
    mForT (R t u) = FFOL.R M (mTmT t) (mTmT u)
    mForT (A ⇒ B) = FFOL._⇒_ M (mForT A) (mForT B)
    mForT {Γ} (∀∀ A) = FFOL.∀∀ M (mForT A)

    mSubt : {Δₜ : Cont}{Γₜ : Cont} → Subt Δₜ Γₜ → (FFOL.Sub M (mCont Δₜ) (mCont Γₜ))
    mSubt εₜ = FFOL.ε M
    mSubt (σₜ ,ₜ t) = FFOL._,ₜ_ M (mSubt σₜ) (mTmT t)




    mConp : {Γₜ : Cont} → Conp Γₜ → (FFOL.Con M)
    mForP : {Γₜ : Cont} {Γₚ : Conp Γₜ} → (For Γₜ) → (FFOL.For M (mConp Γₚ))
    mConp {Γₜ} ◇p = mCont Γₜ
    mConp {Γₜ} (Γₚ ▹p⁰ A) = FFOL._▹ₚ_ M (mConp Γₚ) (mForP {Γₚ = Γₚ} A)
    mForP {Γₜ} {Γₚ = ◇p} A = mForT {Γₜ} A
    mForP {Γₚ = Γₚ ▹p⁰ B} A = FFOL._[_]f M (mForP {Γₚ = Γₚ} A) (FFOL.πₚ¹ M (FFOL.id M))
    
    mTmP : {Γₜ : Cont}{Γₚ : Conp Γₜ} → Tm Γₜ → (FFOL.Tm M (mConp Γₚ))
    mTmP {Γₜ}{Γₚ = ◇p} t = mTmT {Γₜ} t
    mTmP {Γₚ = Γₚ ▹p⁰ x} t = FFOL._[_]t M (mTmP {Γₚ = Γₚ} t) (FFOL.πₚ¹ M (FFOL.id M))

    mCon : Con → (FFOL.Con M)
    mCon Γ = mConp {Con.t Γ} (Con.p Γ)

    mFor : {Γ : Con} → (For (Con.t Γ)) → (FFOL.For M (mCon Γ))
    mFor {Γ} A = mForP {Con.t Γ} {Con.p Γ} A

    mTm : {Γ : Con} → Tm (Con.t Γ) → (FFOL.Tm M (mCon Γ))
    mTm {Γ} t = mTmP {Con.t Γ} {Con.p Γ} t

    e▹ₜT : {Γₜ : Cont} → mCont (Γₜ ▹t⁰) ≡ FFOL._▹ₜ M (mCont Γₜ)
    e▹ₜT = refl
    e▹ₜP : {Γₜ : Cont}{Γₚ : Conp Γₜ} → mConp {Γₜ ▹t⁰} (Γₚ [ wkₜσₜ idₜ ]c) ≡ FFOL._▹ₜ M (mConp Γₚ)
    e▹ₜP {Γₜ = Γₜ} {Γₚ = ◇p} = e▹ₜT {Γₜ = Γₜ}
    e▹ₜP {Γₚ = Γₚ ▹p⁰ A} = {!!}
    e▹ₜ : {Γ : Con} → mCon (con (Con.t Γ ▹t⁰) (Con.p Γ [ wkₜσₜ idₜ ]c)) ≡ FFOL._▹ₜ M (mCon Γ)
    e▹ₜ {Γ} = e▹ₜP {Γₜ = Con.t Γ} {Γₚ = Con.p Γ}

    mForT⇒ : {Γₜ : Cont}{A B : For Γₜ} → mForT {Γₜ} (A ⇒ B) ≡ FFOL._⇒_ M (mForT {Γₜ} A) (mForT {Γₜ} B)
    mForT⇒ = refl
    mForP⇒ : {Γₜ : Cont}{Γₚ : Conp Γₜ}{A B : For Γₜ} → mForP {Γₜ} {Γₚ} (A ⇒ B) ≡ FFOL._⇒_ M (mForP {Γₜ} {Γₚ} A) (mForP {Γₜ} {Γₚ} B)
    mForP⇒ {Γₜ} {Γₚ = ◇p}{A}{B} = mForT⇒ {Γₜ}{A}{B}
    mForP⇒ {Γₚ = Γₚ ▹p⁰ C}{A}{B} = ≡tran (cong (λ X → (M FFOL.[ X ]f) _) (mForP⇒ {Γₚ = Γₚ})) (FFOL.[]f-⇒ M {F = mForP {Γₚ = Γₚ} A} {G = mForP {Γₚ = Γₚ} B} {σ = (FFOL.πₚ¹ M (FFOL.id M))})
    mFor⇒ : {Γ : Con}{A B : For (Con.t Γ)} → mFor {Γ} (A ⇒ B) ≡ FFOL._⇒_ M (mFor {Γ} A) (mFor {Γ} B)
    mFor⇒ {Γ} = mForP⇒ {Con.t Γ} {Con.p Γ}

    mForT∀∀ : {Γₜ : Cont}{A : For (Γₜ ▹t⁰)} → mForT {Γₜ} (∀∀ A) ≡ FFOL.∀∀ M (mForT {Γₜ ▹t⁰} A)
    mForT∀∀ = refl
    mForP∀∀ : {Γₜ : Cont}{Γₚ : Conp Γₜ}{A : For (Γₜ ▹t⁰)} → mForP {Γₜ} {Γₚ} (∀∀ A) ≡ FFOL.∀∀ M (subst (FFOL.For M) (e▹ₜP {Γₜ} {Γₚ}) (mForP {Γₜ ▹t⁰} {Γₚ ▹tp} A))
    -- mFor∀∀ : {Γ : Con}{A : For ((Con.t Γ) ▹t⁰)} → mFor {Γ} (∀∀ A) ≡ FFOL.∀∀ M (mFor {Γ ▹t} A)

    --mForL : {Γ : Con}{A : For (Con.t Γ ▹t⁰)}{t : Tm (Con.t Γ)} → FFOL._[_]f M (mFor {Γ = {!Γ ▹t!}} A) (FFOL._,ₜ_ M (FFOL.id M) (mTm {Γ = Γ} t)) ≡ mFor {Γ = Γ} (A [ idₜ ,ₜ t ]f)

    m⊢ : {Γ : Con} {A : For (Con.t Γ)} → Pf (Con.t Γ) (Con.p Γ) A → FFOL._⊢_ M (mCon Γ) (mFor {Γ = Γ} A)
    m⊢ (var pvzero) = FFOL.πₚ² M (FFOL.id M)
    m⊢ (var (pvnext pv)) = FFOL._[_]p M (m⊢ (var pv)) (FFOL.πₚ¹ M (FFOL.id M))
    m⊢ {Γ} {B} (app {A = A} pf pf') = FFOL.app M (substP (FFOL._⊢_ M _) (mFor⇒ {Γ}{A}{B}) (m⊢ pf)) (m⊢ pf')
    m⊢ {Γ} {A ⇒ B} (lam pf) = substP (FFOL._⊢_ M _) (≡sym (mFor⇒ {Γ}{A}{B})) (FFOL.lam M (m⊢ pf))
    m⊢ {Γ} (p∀∀e {A = A} {t = t} pf) = substP (FFOL._⊢_ M _) {!!} (FFOL.∀e M {F = mFor {{!!}} A} (substP (FFOL._⊢_ M _) {!!} (m⊢ pf)) {t = mTm {Γ} t})
    m⊢ {Γ} (p∀∀i {A = A} pf) = substP (FFOL._⊢_ M _) (≡sym mForP∀∀) (FFOL.∀i M (substP (λ Ξ → FFOL._⊢_ M Ξ (mFor A)) e▹ₜ (m⊢ pf)))


    mSubp : {Γₜ : Cont}{Δₚ Γₚ : Conp Γₜ} → Subp {Γₜ} Δₚ Γₚ → (FFOL.Sub M (mConp Δₚ) (mConp Γₚ))
    mSubp {Γₜ} {Γₚ = ◇p} σₚ = {!FFOL.ε M!}
    mSubp {Γₚ = Γₚ ▹p⁰ A} σₚ = FFOL._,ₚ_ M (mSubp (πₚ¹ σₚ)) {!m⊢ (πₚ² σₚ)!}
    

    mSub : {Δ : Con}{Γ : Con} → Sub Δ Γ → (FFOL.Sub M (mCon Δ) (mCon Γ))
    mSub {Δ}{Γ} σ = FFOL._∘_ M (subst (FFOL.Sub M (mCont (Con.t Δ))) {!!} (mSubt (Sub.t σ))) ({!mSubp (Sub.p σ)!})

    
    e▹ₚ : {Γ : Con}{A : For (Con.t Γ)} → mCon (Γ ▹p A) ≡ FFOL._▹ₚ_ M (mCon Γ) (mFor {Γ} A)
    e[]f : {Γ Δ : Con}{A : For (Con.t Γ)}{σ : Sub Δ Γ} → mFor {Δ} (A [ Sub.t σ ]f) ≡ FFOL._[_]f M (mFor {Γ} A) (mSub σ)
    

    {-
    e▹ₜ {con Γₜ ◇p} = refl
    e▹ₜ {con Γₜ (Γₚ ▹p⁰ A)} = ≡tran²
      (cong₂' (FFOL._▹ₚ_ M) (e▹ₜ {con Γₜ Γₚ}) (cong (subst (FFOL.For M) (e▹ₜ {Γ = con Γₜ Γₚ})) (e[]f {A = A}{σ = πₜ¹* id})))
      (substP (λ X → (M FFOL.▹ₚ (M FFOL.▹ₜ) (mCon (con Γₜ Γₚ))) X ≡ (M FFOL.▹ₜ) ((M FFOL.▹ₚ mCon (con Γₜ Γₚ)) (mFor A)))
        (≡tran
          (coeshift {!!})
          (cong (λ X → subst (FFOL.For M) _ (FFOL._[_]f M (mFor A) (mSub (sub (wkₜσₜ idₜ) X)))) (≡sym (coecoe-coe {eq1 = {!!}} {x = idₚ {Δₚ = Γₚ}}))))
        {!!})
      (cong (M FFOL.▹ₜ) (≡sym (e▹ₚ {con Γₜ Γₚ})))
    -- substP (λ X → FFOL._▹ₚ_ M X (mFor {Γ = ?} (A [ wkₜσₜ idₜ ]f)) ≡ (FFOL._▹ₜ M (mCon (con Γₜ (Γₚ ▹p⁰ A))))) (≡sym (e▹ₜ {Γ = con Γₜ Γₚ})) ?
    -}
    
    
    e[]f = {!!}
    e▹ₚ = {!!}


    {-
    


    

    e∘ : {Γ Δ Ξ : Con}{δ : Sub Δ Ξ}{σ : Sub Γ Δ} → mSub (δ ∘ σ) ≡ FFOL._∘_ M (mSub δ) (mSub σ)
    e∘ = {!!}
    eid : {Γ : Con} → mSub (id {Γ}) ≡ FFOL.id M {mCon Γ}
    eid = {!!}
    e◇ : mCon ◇ ≡ FFOL.◇ M
    e◇ = {!!}
    eε : {Γ : Con} → mSub (sub (εₜ {Con.t Γ}) (εₚ {Con.t Γ} {Con.p Γ})) ≡ subst (FFOL.Sub M (mCon Γ)) (≡sym e◇) (FFOL.ε M {mCon Γ})
    eε = {!!}
    e[]t : {Γ Δ : Con}{t : Tm (Con.t Γ)}{σ : Sub Δ Γ} → mTm (t [ Sub.t σ ]t) ≡ FFOL._[_]t M (mTm t) (mSub σ)
    e[]t = {!!}
    eπₜ¹ : {Γ Δ : Con}{σ : Sub Δ (Γ ▹t)} → mSub (πₜ¹* σ) ≡ FFOL.πₜ¹ M (subst (FFOL.Sub M (mCon Δ)) e▹ₜ (mSub σ))
    eπₜ¹ = {!!}
    eπₜ² : {Γ Δ : Con}{σ : Sub Δ (Γ ▹t)} → mTm (πₜ²* σ) ≡ FFOL.πₜ² M (subst (FFOL.Sub M (mCon Δ)) e▹ₜ (mSub σ))
    eπₜ² = {!!}
    e,ₜ : {Γ Δ : Con}{σ : Sub Δ Γ}{t : Tm (Con.t Δ)} → mSub (σ ,ₜ* t) ≡ subst (FFOL.Sub M (mCon Δ)) (≡sym e▹ₜ) (FFOL._,ₜ_ M (mSub σ) (mTm t))
    e,ₜ = {!!}
    -- Proofs are in prop, so no equation needed
    --[]p : {Γ Δ : Con}{A : For Γ}{pf : FFOL._⊢_ S Γ A}{σ : FFOL.Sub S Δ Γ} → m⊢ (FFOL._[_]p S pf σ) ≡ FFOL._[_]p M (m⊢ pf) (mSub σ)
    e▹ₚ = {!!}
    eπₚ¹ : {Γ Δ : Con}{A : For (Con.t Γ)}{σ : Sub Δ (Γ ▹p A)} → mSub (πₚ¹* σ) ≡ FFOL.πₚ¹ M (subst (FFOL.Sub M (mCon Δ)) e▹ₚ (mSub σ))
    eπₚ¹ = {!!}
    --πₚ² : {Γ Δ : Con}{A : For Γ}{σ : Sub Δ (Γ ▹p A)} → m⊢ (πₚ²* σ) ≡ FFOL.πₚ¹ M (subst (FFOL.Sub M (mCon Δ)) e▹ₚ (mSub σ))
    e,ₚ : {Γ Δ : Con}{A : For (Con.t Γ)}{σ : Sub Δ Γ}{pf : Pf (Con.t Δ) (Con.p Δ) (A [ Sub.t σ ]f)}
      → mSub (σ ,ₚ* pf) ≡ subst (FFOL.Sub M (mCon Δ)) (≡sym e▹ₚ) (FFOL._,ₚ_ M (mSub σ) (substP (FFOL._⊢_ M (mCon Δ)) e[]f (m⊢ pf)))
    e,ₚ = {!!}
    eR : {Γ : Con}{t u : Tm (Con.t Γ)} → mFor (R t u) ≡ FFOL.R M (mTm t) (mTm u)
    eR = {!!}
    e⇒ : {Γ : Con}{A B : For (Con.t Γ)} → mFor (A ⇒ B) ≡ FFOL._⇒_ M (mFor A) (mFor B)
    e⇒ = {!!}
    e∀∀ : {Γ : Con}{A : For ((Con.t Γ) ▹t⁰)} → mFor (∀∀ A) ≡ FFOL.∀∀ M (subst (FFOL.For M) e▹ₜ (mFor A))
    e∀∀ = {!!}
    -}
    m : Mapping ffol M
    m = record { mCon = mCon ; mSub = mSub ; mTm = λ {Γ} t → mTm {Γ} t ; mFor = λ {Γ} A → mFor {Γ} A ; m⊢ = m⊢ }
    
  --mor : (M : FFOL) → Morphism ffol M
  --mor M = record {InitialMorphism M}

  -}
  
\end{code}