From 6829fe86c7a85abbeb196e9d3ce5484de79a2fb8 Mon Sep 17 00:00:00 2001 From: Mysaa Date: Thu, 20 Jul 2023 15:48:47 +0200 Subject: [PATCH] Started tidying up the syntax --- FFOL.agda | 25 +++-- FFOLInitial.agda | 247 ++++++++++++++++++++++++++--------------------- 2 files changed, 152 insertions(+), 120 deletions(-) diff --git a/FFOL.agda b/FFOL.agda index 9e81cea..6baeb69 100644 --- a/FFOL.agda +++ b/FFOL.agda @@ -14,6 +14,8 @@ module FFOL where infixr 10 _∘_ infixr 5 _⊢_ field + + -- We first make the base category with its final object Con : Set ℓ¹ Sub : Con → Con → Set ℓ⁵ -- It makes a category _∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ @@ -21,8 +23,8 @@ module FFOL where id : {Γ : Con} → Sub Γ Γ idl : {Γ Δ : Con} {σ : Sub Γ Δ} → (id {Δ}) ∘ σ ≡ σ idr : {Γ Δ : Con} {σ : Sub Γ Δ} → σ ∘ (id {Γ}) ≡ σ - ◇ : Con -- The initial object of the category - ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object + ◇ : Con -- The terminal object of the category + ε : {Γ : Con} → Sub Γ ◇ -- The morphism from any object to the terminal ε-u : {Γ : Con} → {σ : Sub Γ ◇} → σ ≡ ε {Γ} -- Functor Con → Set called Tm @@ -31,7 +33,7 @@ module FFOL where []t-id : {Γ : Con} → {x : Tm Γ} → x [ id {Γ} ]t ≡ x []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 - -- Tm : Set+ + -- Tm : Set⁺ _▹ₜ : Con → Con πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ @@ -47,11 +49,7 @@ module FFOL where []f-id : {Γ : Con} → {F : For Γ} → F [ id {Γ} ]f ≡ F []f-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → F [ β ∘ α ]f ≡ (F [ β ]f) [ α ]f - -- Formulas with relation on terms - R : {Γ : Con} → (t u : Tm Γ) → For Γ - R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t) - - -- Proofs + -- Functor Con × For → Prop called Pf or ⊢ _⊢_ : (Γ : Con) → For Γ → Prop ℓ⁴ _[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms -- Equalities below are useless because Γ ⊢ F is in prop @@ -63,12 +61,17 @@ module FFOL where πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f) _,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F) - -- Equalities below are useless because Γ ⊢ F is in Prop ,ₚ∘πₚ : {Γ Δ : Con} → {F : For Γ} → {σ : Sub Δ (Γ ▹ₚ F)} → (πₚ¹ σ) ,ₚ (πₚ² σ) ≡ σ πₚ¹∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ¹ (σ ,ₚ prf) ≡ σ + -- Equality below is useless because Γ ⊢ F is in Prop -- πₚ²∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ² (σ ,ₚ prf) ≡ prf ,ₚ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{F : For Ξ}{prf : Γ ⊢ (F [ σ ]f)} → (σ ,ₚ prf) ∘ δ ≡ (σ ∘ δ) ,ₚ (substP (λ F → Δ ⊢ F) (≡sym []f-∘) (prf [ δ ]p)) + + {-- FORMULAE CONSTRUCTORS --} + -- Formulas with relation on terms + R : {Γ : Con} → (t u : Tm Γ) → For Γ + R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t) -- Implication _⇒_ : {Γ : Con} → For Γ → For Γ → For Γ @@ -78,10 +81,12 @@ module FFOL where ∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ []f-∀∀ : {Γ Δ : Con} → {F : For (Γ ▹ₜ)} → {σ : Sub Δ Γ} → (∀∀ F) [ σ ]f ≡ (∀∀ (F [ (σ ∘ πₜ¹ id) ,ₜ πₜ² id ]f)) + {-- PROOFS CONSTRUCTORS --} + -- Again, we don't have to write the _[_]p equalities as Proofs are in Prop + -- Lam & App lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G) app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G - -- Again, we don't write the _[_]p equalities as everything is in Prop -- ∀i and ∀e ∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F) diff --git a/FFOLInitial.agda b/FFOLInitial.agda index 902ebd6..5996475 100644 --- a/FFOLInitial.agda +++ b/FFOLInitial.agda @@ -8,78 +8,44 @@ module FFOLInitial where open import Agda.Primitive open import ListUtil - -- First definition of terms and term contexts -- + {-- 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 + + -- A term variable is a de-bruijn variable, TmVar n ≈ ⟦0,n-1⟧ data TmVar : Cont → Set₁ where tvzero : TmVar (Γₜ ▹t⁰) tvnext : TmVar Γₜ → TmVar (Γₜ ▹t⁰) - + + -- For now, we only have term variables (no function symbol) data Tm : Cont → Set₁ where var : TmVar Γₜ → Tm Γₜ -- Now we can define formulæ data For : Cont → Set₁ where - r : Tm Γₜ → Tm Γₜ → For Γₜ + R : Tm Γₜ → Tm Γₜ → For Γₜ _⇒_ : For Γₜ → For Γₜ → For Γₜ ∀∀ : For (Γₜ ▹t⁰) → For Γₜ - -- Then we define term substitutions, and the application of them on terms and formulæ + + + + -- Then we define term substitutions data Subt : Cont → Cont → Set₁ where εₜ : Subt Γₜ ◇t _,ₜ_ : Subt Δₜ Γₜ → Tm Δₜ → Subt Δₜ (Γₜ ▹t⁰) - -- We subst on terms - _[_]t : Tm Γₜ → Subt Δₜ Γₜ → Tm Δₜ - var tvzero [ σ ,ₜ t ]t = t - var (tvnext tv) [ σ ,ₜ t ]t = var tv [ σ ]t - - -- We define liftings on term variables - -- A term of n variables is a term of n+1 variables - -- Same for a term array - 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) - wkₜσt-wkₜt : {tv : TmVar Γₜ} → {σ : Subt Δₜ Γₜ} → wkₜt (var tv [ σ ]t) ≡ var tv [ wkₜσt σ ]t - wkₜσt-wkₜt {tv = tvzero} {σ = σ ,ₜ x} = refl - wkₜσt-wkₜt {tv = tvnext tv} {σ = σ ,ₜ x} = wkₜσt-wkₜt {tv = tv} {σ = σ} - - -- 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 - liftₜσ : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) (Γₜ ▹t⁰) - liftₜσ σ = (wkₜσt σ) ,ₜ (var tvzero) - - - -- We subst on formulæ - _[_]f : For Γₜ → Subt Δₜ Γₜ → For Δₜ - (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 [ liftₜσ σ ]f) - - -- We now can define identity on term substitutions - idₜ : Subt Γₜ Γₜ - idₜ {◇t} = εₜ - idₜ {Γₜ ▹t⁰} = liftₜσ (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 have the access functions from the algebra, in restricted versions + -- We write down the access functions from the algebra, in restricted versions πₜ¹ : Subt Δₜ (Γₜ ▹t⁰) → Subt Δₜ Γₜ πₜ¹ (σₜ ,ₜ t) = σₜ πₜ² : Subt Δₜ (Γₜ ▹t⁰) → Tm Δₜ πₜ² (σₜ ,ₜ t) = t - -- 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 @@ -88,64 +54,125 @@ module FFOLInitial where ,ₜ∘πₜ : {σₜ : Subt Δₜ (Γₜ ▹t⁰)} → (πₜ¹ σₜ) ,ₜ (πₜ² σₜ) ≡ σₜ ,ₜ∘πₜ {σₜ = σₜ ,ₜ t} = refl - -- We can also prove the substitution equalities + + -- We now define the action of term substitutions on terms + _[_]t : Tm Γₜ → Subt Δₜ Γₜ → Tm Δₜ + var tvzero [ σ ,ₜ t ]t = t + var (tvnext tv) [ σ ,ₜ t ]t = var tv [ σ ]t + + -- 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 + _[_]f : For Γₜ → Subt Δₜ Γₜ → For Δₜ + (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 [ lfₜσₜ σ ]f) + + + + + -- 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)) (wkₜσt-wkₜt {tv = tv} {σ = idₜ}) (substP (λ t → wkₜt t ≡ var (tvnext tv)) (≡sym ([]t-id {t = var tv})) 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} - []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 - wkₜσt-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → wkₜσt (β ∘ₜ α) ≡ (wkₜσt β ∘ₜ liftₜσ α) - wkₜt[] : {α : Subt Δₜ Γₜ} → {t : Tm Γₜ} → wkₜt (t [ α ]t) ≡ (wkₜt t [ liftₜσ α ]t) - wkₜσt-∘ {β = εₜ} = refl - wkₜσt-∘ {β = β ,ₜ t} = cong₂ _,ₜ_ wkₜσt-∘ (wkₜt[] {t = t}) - wkₜt[] {α = α ,ₜ t} {var tvzero} = refl - wkₜt[] {α = α ,ₜ t} {var (tvnext tv)} = wkₜt[] {t = var tv} - liftₜσ-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → liftₜσ (β ∘ₜ α) ≡ (liftₜσ β) ∘ₜ (liftₜσ α) - liftₜσ-∘ {α = α} {β = εₜ} = refl - liftₜσ-∘ {α = α} {β = β ,ₜ t} = cong₂ _,ₜ_ (cong₂ _,ₜ_ wkₜσt-∘ (wkₜt[] {t = t})) refl - []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) liftₜσ-∘) []f-∘) - R[] : {σ : Subt Δₜ Γₜ} → {t u : Tm Γₜ} → (r t u) [ σ ]f ≡ r (t [ σ ]t) (u [ σ ]t) - R[] = refl - lem3 : {α : Subt Γₜ Δₜ} → {β : Subt Ξₜ Γₜ} → α ∘ₜ (wkₜσt β) ≡ wkₜσt (α ∘ₜ β) - lem3 {α = εₜ} = refl - lem3 {α = α ,ₜ var tv} = cong₂ _,ₜ_ (lem3 {α = α}) (≡sym (wkₜσt-wkₜt {tv = tv})) - wk[,] : {t : Tm Γₜ}{u : Tm Δₜ}{β : Subt Δₜ Γₜ} → (wkₜt t) [ β ,ₜ u ]t ≡ t [ β ]t - wk[,] {t = var tvzero} = refl - wk[,] {t = var (tvnext tv)} = refl - wk∘, : {α : Subt Γₜ Δₜ}{β : Subt Ξₜ Γₜ}{t : Tm Ξₜ} → (wkₜσt α) ∘ₜ (β ,ₜ t) ≡ (α ∘ₜ β) - wk∘, {α = εₜ} = refl - wk∘, {α = α ,ₜ t} {β = β} = cong₂ _,ₜ_ (wk∘, {α = α}) (wk[,] {t = t} {β = β}) - σ-idl : {α : Subt Δₜ Γₜ} → idₜ ∘ₜ α ≡ α - σ-idl {α = εₜ} = refl - σ-idl {α = α ,ₜ x} = cong₂ _,ₜ_ (≡tran wk∘, σ-idl) refl - σ-idr : {α : Subt Δₜ Γₜ} → α ∘ₜ idₜ ≡ α - σ-idr {α = εₜ} = refl - σ-idr {α = α ,ₜ x} = cong₂ _,ₜ_ σ-idr []t-id + + -- 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})) - []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 σ-idr)) (≡sym lem3)) refl)) + + -- Unicity of the terminal morphism εₜ-u : {σₜ : Subt Γₜ ◇t} → σₜ ≡ εₜ εₜ-u {σₜ = εₜ} = refl - data Conp : Cont → Set₁ -- pu tit in Prop + -- 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 : Conp Ξₜ - - data Conp where - ◇p : Conp Γₜ - _▹p⁰_ : Conp Γₜ → For Γₜ → Conp Γₜ record Con : Set₁ where constructor con @@ -166,10 +193,10 @@ module FFOLInitial where -- We can add term, that will not be used in the formulæ already present - -- (that's why we use wkₜσt) + -- (that's why we use wkₜσₜ) _▹tp : Conp Γₜ → Conp (Γₜ ▹t⁰) ◇p ▹tp = ◇p - (Γₚ ▹p⁰ A) ▹tp = (Γₚ ▹tp) ▹p⁰ (A [ wkₜσt idₜ ]f) + (Γₚ ▹p⁰ A) ▹tp = (Γₚ ▹tp) ▹p⁰ (A [ wkₜσₜ idₜ ]f) _▹t : Con → Con Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp) @@ -182,7 +209,7 @@ module FFOLInitial where var : {A : For (Con.t Γ)} → PfVar Γ A → Pf Γ A app : {A B : For (Con.t Γ)} → Pf Γ (A ⇒ B) → Pf Γ A → Pf Γ B lam : {A B : For (Con.t Γ)} → Pf (Γ ▹p A) B → Pf Γ (A ⇒ B) - p∀∀e : {A : For ((Con.t Γ) ▹t⁰)} → {t : Tm (Con.t Γ)} → Pf Γ (∀∀ A) → Pf Γ (A [ idₜ ,ₜ t ]f) + p∀∀e : {A : For (Con.t (Γ ▹t))} → {t : Tm (Con.t Γ)} → Pf Γ (∀∀ A) → Pf Γ (A [ idₜ ,ₜ t ]f) p∀∀i : {A : For (Con.t (Γ ▹t))} → Pf (Γ ▹t) A → Pf Γ (∀∀ A) @@ -226,7 +253,7 @@ module FFOLInitial where refl∈ₚ* {Γₚ = ◇p} = zero∈ₚ* refl∈ₚ* {Γₚ = Γₚ ▹p⁰ x} = both∈ₚ* refl∈ₚ* - ∈ₚ▹tp : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → PfVar (con Δₜ Δₚ ▹t) (A [ wkₜσt idₜ ]f) + ∈ₚ▹tp : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → PfVar (con Δₜ Δₚ ▹t) (A [ wkₜσₜ idₜ ]f) ∈ₚ▹tp pvzero = pvzero ∈ₚ▹tp (pvnext x) = pvnext (∈ₚ▹tp x) ∈ₚ*▹tp : Γₚ ∈ₚ* Δₚ → (Γₚ ▹tp) ∈ₚ* (Δₚ ▹tp) @@ -255,9 +282,9 @@ module FFOLInitial where wkₚσt : {Δₜ : Cont} {Δₚ Γₚ : Conp Δₜ}{A : For Δₜ} → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} (Δₚ ▹p⁰ A) Γₚ wkₚσt εₚ = εₚ wkₚσt (σₚ ,ₚ pf) = (wkₚσt σₚ) ,ₚ wkₚp (right∈ₚ* refl∈ₚ*) pf - --wkₜσt-wkₜt : {tv : TmVar Γₜ} → {σ : Subt Δₜ Γₜ} → wkₜt (var tv [ σ ]t) ≡ var tv [ wkₜσt σ ]t - --wkₜσt-wkₜt {tv = tvzero} {σ = σ ,ₜ x} = refl - --wkₜσt-wkₜt {tv = tvnext tv} {σ = σ ,ₜ x} = wkₜσt-wkₜt {tv = tv} {σ = σ} + --wkₜt-wkₜσₜ : {tv : TmVar Γₜ} → {σ : Subt Δₜ Γₜ} → wkₜt (var tv [ σ ]t) ≡ var tv [ wkₜσₜ σ ]t + --wkₜt-wkₜσₜ {tv = tvzero} {σ = σ ,ₜ x} = refl + --wkₜt-wkₜσₜ {tv = tvnext tv} {σ = σ ,ₜ x} = wkₜt-wkₜσₜ {tv = tv} {σ = σ} -- 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 @@ -276,11 +303,11 @@ module FFOLInitial where - lem7 : {σ : Subt Δₜ Γₜ} → ((Δₚ ▹tp) [ liftₜσ σ ]c) ≡ ((Δₚ [ σ ]c) ▹tp) + lem7 : {σ : Subt Δₜ Γₜ} → ((Δₚ ▹tp) [ lfₜσₜ σ ]c) ≡ ((Δₚ [ σ ]c) ▹tp) lem7 {Δₚ = ◇p} = refl - lem7 {Δₚ = Δₚ ▹p⁰ A} = cong₂ _▹p⁰_ lem7 (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (≡tran² (≡sym wkₜσt-∘) (cong wkₜσt (≡tran σ-idl (≡sym σ-idr))) (≡sym lem3))) []f-∘) - lem8 : {σ : Subt Δₜ Γₜ} {t : Tm Γₜ} → ((wkₜσt σ ∘ₜ (idₜ ,ₜ (t [ σ ]t))) ,ₜ (t [ σ ]t)) ≡ ((idₜ ∘ₜ σ) ,ₜ (t [ σ ]t)) - lem8 = cong₂ _,ₜ_ (≡tran² wk∘, σ-idr (≡sym σ-idl)) refl + lem7 {Δₚ = Δₚ ▹p⁰ A} = cong₂ _▹p⁰_ lem7 (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (≡tran² (≡sym wkₜσₜ-∘ₜl) (cong wkₜσₜ (≡tran idlₜ (≡sym idrₜ))) (≡sym wkₜσₜ-∘ₜr))) []f-∘) + lem8 : {σ : Subt Δₜ Γₜ} {t : Tm Γₜ} → ((wkₜσₜ σ ∘ₜ (idₜ ,ₜ (t [ σ ]t))) ,ₜ (t [ σ ]t)) ≡ ((idₜ ∘ₜ σ) ,ₜ (t [ σ ]t)) + lem8 = cong₂ _,ₜ_ (≡tran² wkₜ∘ₜ,ₜ idrₜ (≡sym idlₜ)) refl _[_]pvₜ : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → (σ : Subt Γₜ Δₜ) → PfVar (con Γₜ (Δₚ [ σ ]c)) (A [ σ ]f) _[_]pₜ : {A : For Δₜ} → Pf (con Δₜ Δₚ) A → (σ : Subt Γₜ Δₜ) → Pf (con Γₜ (Δₚ [ σ ]c)) (A [ σ ]f) @@ -290,18 +317,18 @@ module FFOLInitial where 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 (con Γₜ (Δₚ [ σ ]c)) F) (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (lem8 {t = t})) ([]f-∘)) (p∀∀e {t = t [ σ ]t} (pf [ σ ]pₜ)) - _[_]pₜ {Γₜ = Γₜ} (p∀∀i pf) σ = p∀∀i (substP (λ Ξₚ → Pf (con (Γₜ ▹t⁰) (Ξₚ)) _) lem7 (pf [ liftₜσ σ ]pₜ)) + _[_]pₜ {Γₜ = Γₜ} (p∀∀i pf) σ = p∀∀i (substP (λ Ξₚ → Pf (con (Γₜ ▹t⁰) (Ξₚ)) _) lem7 (pf [ lfₜσₜ σ ]pₜ)) _[_]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) - lem9 : (Δₚ [ wkₜσt idₜ ]c) ≡ (Δₚ ▹tp) + lem9 : (Δₚ [ wkₜσₜ idₜ ]c) ≡ (Δₚ ▹tp) lem9 {Δₚ = ◇p} = refl lem9 {Δₚ = Δₚ ▹p⁰ x} = cong₂ _▹p⁰_ lem9 refl 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 (con (Δₜ ▹t⁰) Ξₚ) (A [ wkₜσt idₜ ]f)) lem9 (_[_]pₜ {Γₜ = Δₜ ▹t⁰} pf (wkₜσt idₜ)) + wkₜσₚ {Δₜ = Δₜ} (_,ₚ_ {A = A} σₚ pf) = (wkₜσₚ σₚ) ,ₚ substP (λ Ξₚ → Pf (con (Δₜ ▹t⁰) Ξₚ) (A [ wkₜσₜ idₜ ]f)) lem9 (_[_]pₜ {Γₜ = Δₜ ▹t⁰} pf (wkₜσₜ idₜ)) _[_]p : {A : For Δₜ} → Pf (con Δₜ Δₚ) A → (σ : Subp {Δₜ} Δₚ' Δₚ) → Pf (con Δₜ Δₚ') A var pvzero [ σ ,ₚ pf ]p = pf @@ -362,7 +389,7 @@ module FFOLInitial where _∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ sub αₜ αₚ ∘ sub βₜ βₚ = sub (αₜ ∘ₜ βₜ) (subst (Subp _) (≡sym []c-∘) (αₚ [ βₜ ]σₚ) ∘ₚ βₚ) idl : {Γ Δ : Con} {σ : Sub Γ Δ} → (id {Δ}) ∘ σ ≡ σ - idl {Δ = Δ} {σ = sub σₜ σₚ} = cong₂' sub σ-idl (≡tran (substfpoly {α = ((Con.p Δ) [ idₜ ∘ₜ σₜ ]c)} {β = ((Con.p Δ) [ σₜ ]c)} {eq = cong (λ ξ → Con.p Δ [ ξ ]c) σ-idl} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ σₚ}) (≡tran (cong₂ _∘ₚ_ (≡tran³ coecoe-coe (substfpoly {eq = []c-id} {f = λ {Ξₚ} ξ → _[_]σₚ {Δₚ = Con.p Δ} {Δₚ' = Ξₚ} ξ σₜ}) (cong (λ ξ → ξ [ σₜ ]σₚ) coeaba) idₚ[]) refl) idlₚ)) + idl {Δ = Δ} {σ = sub σₜ σₚ} = cong₂' sub idlₜ (≡tran (substfpoly {α = ((Con.p Δ) [ idₜ ∘ₜ σₜ ]c)} {β = ((Con.p Δ) [ σₜ ]c)} {eq = cong (λ ξ → Con.p Δ [ ξ ]c) idlₜ} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ σₚ}) (≡tran (cong₂ _∘ₚ_ (≡tran³ coecoe-coe (substfpoly {eq = []c-id} {f = λ {Ξₚ} ξ → _[_]σₚ {Δₚ = Con.p Δ} {Δₚ' = Ξₚ} ξ σₜ}) (cong (λ ξ → ξ [ σₜ ]σₚ) coeaba) idₚ[]) refl) idlₚ)) lemK : {Γ Δ : Con}{σₜ : Subt (Con.t Γ) (Con.t Δ)}{σₚ : Subp (Con.p Γ [ idₜ ]c) ((Con.p Δ [ σₜ ]c)[ idₜ ]c)} {eq1 : Subp (Con.p Γ) ((Con.p Δ [ σₜ ]c) [ idₜ ]c) ≡ Subp (Con.p Γ) (Con.p Δ [ σₜ ]c)} {eq2 : Subp (Con.p Γ) (Con.p Γ) ≡ Subp (Con.p Γ) (Con.p Γ [ idₜ ]c)} @@ -371,7 +398,7 @@ module FFOLInitial where ≡ (coe eq3 σₚ ∘ₚ idₚ) lemK {Γ}{Δ}{σₚ = σₚ}{eq1}{eq2}{eq3} = substP (λ X → coe eq1 (X ∘ₚ coe eq2 idₚ) ≡ (coe eq3 σₚ ∘ₚ idₚ)) (coeaba {eq1 = eq3}{eq2 = ≡sym eq3}) (coep∘ {p = λ {Γₚ}{Δₚ}{Ξₚ} x y → _∘ₚ_ {Δₚ = Γₚ} x y} {eq1 = refl}{eq2 = ≡sym []c-id}{eq3 = ≡sym []c-id}) idr : {Γ Δ : Con} {σ : Sub Γ Δ} → σ ∘ (id {Γ}) ≡ σ - idr {Γ} {Δ} {σ = sub σₜ σₚ} = cong₂' sub σ-idr (≡tran⁴ (cong (coe _) (≡sym (substfpoly {eq = ≡sym ([]c-∘ {α = σₜ} {β = idₜ}{Ξₚ = Con.p Δ})} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ (coe (cong (Subp (Con.p Γ)) (≡sym []c-id)) idₚ)} {x = σₚ [ idₜ ]σₚ}))) coecoe-coe lemK idrₚ []σₚ-id) + idr {Γ} {Δ} {σ = sub σₜ σₚ} = cong₂' sub idrₜ (≡tran⁴ (cong (coe _) (≡sym (substfpoly {eq = ≡sym ([]c-∘ {α = σₜ} {β = idₜ}{Ξₚ = Con.p Δ})} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ (coe (cong (Subp (Con.p Γ)) (≡sym []c-id)) idₚ)} {x = σₚ [ idₜ ]σₚ}))) coecoe-coe lemK idrₚ []σₚ-id) ∘ₚ-ass : {Γₚ Δₚ Ξₚ Ψₚ : Conp Γₜ}{αₚ : Subp Γₚ Δₚ}{βₚ : Subp Δₚ Ξₚ}{γₚ : Subp Ξₚ Ψₚ} → (γₚ ∘ₚ βₚ) ∘ₚ αₚ ≡ γₚ ∘ₚ (βₚ ∘ₚ αₚ) ∘ₚ-ass {γₚ = εₚ} = refl ∘ₚ-ass {αₚ = αₚ} {βₚ} {γₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ (x [ βₚ ∘ₚ αₚ ]p)) ∘ₚ-ass @@ -432,7 +459,7 @@ module FFOLInitial where lemA : {σₜ : Subt Γₜ Δₜ}{t : Tm Γₜ} → (Γₚ ▹tp) [ σₜ ,ₜ t ]c ≡ Γₚ [ σₜ ]c lemA {Γₚ = ◇p} = refl - lemA {Γₚ = Γₚ ▹p⁰ t} = cong₂ _▹p⁰_ lemA (≡tran (≡sym []f-∘) (cong (λ σ → t [ σ ]f) (≡tran wk∘, σ-idl))) + lemA {Γₚ = Γₚ ▹p⁰ t} = cong₂ _▹p⁰_ lemA (≡tran (≡sym []f-∘) (cong (λ σ → t [ σ ]f) (≡tran wkₜ∘ₜ,ₜ idlₜ))) πₜ¹* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Sub Δ Γ πₜ¹* (sub (σₜ ,ₜ t) σₚ) = sub σₜ (subst (Subp _) lemA σₚ) πₜ²* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Tm (Con.t Δ) @@ -493,7 +520,7 @@ module FFOLInitial where lemC : {σₜ : Subt Δₜ Γₜ}{t : Tm Δₜ} → (Γₚ ▹tp) [ σₜ ,ₜ t ]c ≡ Γₚ [ σₜ ]c lemC {Γₚ = ◇p} = refl - lemC {Γₚ = Γₚ ▹p⁰ x} = cong₂ _▹p⁰_ lemC (≡tran (≡sym []f-∘) (cong (λ σ → x [ σ ]f) (≡tran wk∘, σ-idl))) + lemC {Γₚ = Γₚ ▹p⁰ x} = cong₂ _▹p⁰_ lemC (≡tran (≡sym []f-∘) (cong (λ σ → x [ σ ]f) (≡tran wkₜ∘ₜ,ₜ idlₜ))) lemD : {A : For (Con.t Γ)}{σ : Sub Δ (Γ ▹p A)} → Sub.t (πₚ¹* σ) ≡ Sub.t σ lemD {σ = sub σₜ (σₚ ,ₚ pf)} = refl @@ -527,7 +554,7 @@ module FFOLInitial where ; _[_]f = λ A σ → A [ Sub.t σ ]f ; []f-id = []f-id ; []f-∘ = []f-∘ - ; R = r + ; R = R ; R[] = refl ; _⊢_ = Pf ; _[_]p = λ pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p