More tiding up
This commit is contained in:
parent
6829fe86c7
commit
11a6742677
GPG Key ID:
7054D5D6A90F084F
411
FFOLInitial.agda
411
FFOLInitial.agda
@ -172,185 +172,202 @@ module FFOLInitial where
|
||||
variable
|
||||
Γₚ Γₚ' : Conp Γₜ
|
||||
Δₚ Δₚ' : Conp Δₜ
|
||||
Ξₚ : Conp Ξₜ
|
||||
|
||||
record Con : Set₁ where
|
||||
constructor con
|
||||
field
|
||||
t : Cont
|
||||
p : Conp t
|
||||
|
||||
◇ : Con
|
||||
◇ = con ◇t ◇p
|
||||
|
||||
|
||||
_▹p_ : (Γ : Con) → For (Con.t Γ) → Con
|
||||
Γ ▹p A = con (Con.t Γ) (Con.p Γ ▹p⁰ A)
|
||||
|
||||
variable
|
||||
Γ Δ Ξ : Con
|
||||
|
||||
|
||||
|
||||
-- We can add term, that will not be used in the formulæ already present
|
||||
-- (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)
|
||||
|
||||
_▹t : Con → Con
|
||||
Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp)
|
||||
|
||||
data PfVar : (Γ : Con) → For (Con.t Γ) → Set₁ where
|
||||
pvzero : {A : For (Con.t Γ)} → PfVar (Γ ▹p A) A
|
||||
pvnext : {A B : For (Con.t Γ)} → PfVar Γ A → PfVar (Γ ▹p B) A
|
||||
|
||||
data Pf : (Γ : Con) → For (Con.t Γ) → Prop₁ 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 ▹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)
|
||||
|
||||
|
||||
data Subp : {Δₜ : Cont} → Conp Δₜ → Conp Δₜ → Set₁ where
|
||||
εₚ : Subp Δₚ ◇p
|
||||
_,ₚ_ : {A : For Δₜ} → (σ : Subp Δₚ Δₚ') → Pf (con Δₜ Δₚ) A → Subp Δₚ (Δₚ' ▹p⁰ A)
|
||||
|
||||
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)
|
||||
◇p ▹tp = ◇p
|
||||
(Γₚ ▹p⁰ A) ▹tp = (Γₚ ▹tp) ▹p⁰ (A [ wkₜσₜ idₜ ]f)
|
||||
-- We show how it interacts with ,ₜ and lfₜσₜ
|
||||
lem9 : (Δₚ [ wkₜσₜ idₜ ]c) ≡ (Δₚ ▹tp)
|
||||
lem9 {Δₚ = ◇p} = refl
|
||||
lem9 {Δₚ = Δₚ ▹p⁰ x} = cong₂ _▹p⁰_ lem9 refl
|
||||
▹tp,ₜ : {σₜ : Subt Γₜ Δₜ}{t : Tm Γₜ} → (Γₚ ▹tp) [ σₜ ,ₜ t ]c ≡ Γₚ [ σₜ ]c
|
||||
▹tp,ₜ {Γₚ = ◇p} = refl
|
||||
▹tp,ₜ {Γₚ = Γₚ ▹p⁰ t} = cong₂ _▹p⁰_ ▹tp,ₜ (≡tran (≡sym []f-∘) (cong (λ σ → t [ σ ]f) (≡tran wkₜ∘ₜ,ₜ idlₜ)))
|
||||
▹tp-lfₜ : {σ : Subt Δₜ Γₜ} → ((Δₚ ▹tp) [ lfₜσₜ σ ]c) ≡ ((Δₚ [ σ ]c) ▹tp)
|
||||
▹tp-lfₜ {Δₚ = ◇p} = refl
|
||||
▹tp-lfₜ {Δₚ = Δₚ ▹p⁰ A} = cong₂ _▹p⁰_ ▹tp-lfₜ (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (≡tran² (≡sym wkₜσₜ-∘ₜl) (cong wkₜσₜ (≡tran idlₜ (≡sym idrₜ))) (≡sym wkₜσₜ-∘ₜr))) []f-∘)
|
||||
|
||||
|
||||
|
||||
-- With those contexts, we have everything to define proofs
|
||||
data PfVar : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Set₁ where
|
||||
pvzero : {A : For Γₜ} → PfVar Γₜ (Γₚ ▹p⁰ A) A
|
||||
pvnext : {A B : For Γₜ} → PfVar Γₜ Γₚ A → PfVar Γₜ (Γₚ ▹p⁰ B) A
|
||||
|
||||
data Pf : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Prop₁ where
|
||||
var : {A : For Γₜ} → PfVar Γₜ Γₚ A → Pf Γₜ Γₚ A
|
||||
app : {A B : For Γₜ} → Pf Γₜ Γₚ (A ⇒ B) → Pf Γₜ Γₚ A → Pf Γₜ Γₚ B
|
||||
lam : {A B : For Γₜ} → Pf Γₜ (Γₚ ▹p⁰ A) B → Pf Γₜ Γₚ (A ⇒ B)
|
||||
p∀∀e : {A : For (Γₜ ▹t⁰)} → {t : Tm Γₜ} → Pf Γₜ Γₚ (∀∀ A) → Pf Γₜ Γₚ (A [ idₜ ,ₜ t ]f)
|
||||
p∀∀i : {A : For (Γₜ ▹t⁰)} → Pf (Γₜ ▹t⁰) (Γₚ ▹tp) A → Pf Γₜ Γₚ (∀∀ A)
|
||||
|
||||
|
||||
|
||||
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)
|
||||
|
||||
-- An order on contexts, where we can only change positions
|
||||
infixr 5 _∈ₚ*_
|
||||
data _∈ₚ*_ : Conp Γₜ → Conp Γₜ → Set₁ where
|
||||
zero∈ₚ* : ◇p ∈ₚ* Γₚ
|
||||
next∈ₚ* : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → Δₚ' ∈ₚ* Δₚ → (Δₚ' ▹p⁰ A) ∈ₚ* Δₚ
|
||||
-- Allows to grow ∈ₚ* to the right
|
||||
right∈ₚ* :{A : For Δₜ} → Γₚ ∈ₚ* Δₚ → Γₚ ∈ₚ* (Δₚ ▹p⁰ A)
|
||||
right∈ₚ* zero∈ₚ* = zero∈ₚ*
|
||||
right∈ₚ* (next∈ₚ* x h) = next∈ₚ* (pvnext x) (right∈ₚ* h)
|
||||
both∈ₚ* : {A : For Γₜ} → Γₚ ∈ₚ* Δₚ → (Γₚ ▹p⁰ A) ∈ₚ* (Δₚ ▹p⁰ A)
|
||||
both∈ₚ* zero∈ₚ* = next∈ₚ* pvzero zero∈ₚ*
|
||||
both∈ₚ* (next∈ₚ* x h) = next∈ₚ* pvzero (next∈ₚ* (pvnext x) (right∈ₚ* h))
|
||||
refl∈ₚ* : Γₚ ∈ₚ* Γₚ
|
||||
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 pvzero = pvzero
|
||||
∈ₚ▹tp (pvnext x) = pvnext (∈ₚ▹tp x)
|
||||
∈ₚ*▹tp : Γₚ ∈ₚ* Δₚ → (Γₚ ▹tp) ∈ₚ* (Δₚ ▹tp)
|
||||
∈ₚ*▹tp zero∈ₚ* = zero∈ₚ*
|
||||
∈ₚ*▹tp (next∈ₚ* x s) = next∈ₚ* (∈ₚ▹tp x) (∈ₚ*▹tp s)
|
||||
|
||||
mon∈ₚ∈ₚ* : {A : For Δₜ} → PfVar (con Δₜ Δₚ') A → Δₚ' ∈ₚ* Δₚ → PfVar (con Δₜ Δₚ) A
|
||||
mon∈ₚ∈ₚ* pvzero (next∈ₚ* x x₁) = x
|
||||
mon∈ₚ∈ₚ* (pvnext s) (next∈ₚ* x x₁) = mon∈ₚ∈ₚ* s x₁
|
||||
|
||||
∈ₚ*→Sub : Δₚ ∈ₚ* Δₚ' → Subp {Δₜ} Δₚ' Δₚ
|
||||
∈ₚ*→Sub zero∈ₚ* = εₚ
|
||||
∈ₚ*→Sub (next∈ₚ* x s) = ∈ₚ*→Sub s ,ₚ var x
|
||||
|
||||
|
||||
wkₚp : {A : For Δₜ} → Δₚ ∈ₚ* Δₚ' → Pf (con Δₜ Δₚ) A → Pf (con Δₜ Δₚ') A
|
||||
wkₚp s (var pv) = var (mon∈ₚ∈ₚ* pv s)
|
||||
wkₚp s (app pf pf₁) = app (wkₚp s pf) (wkₚp s pf₁)
|
||||
wkₚp s (lam {A = A} pf) = lam (wkₚp (both∈ₚ* s) pf)
|
||||
wkₚp s (p∀∀e pf) = p∀∀e (wkₚp s pf)
|
||||
wkₚp s (p∀∀i pf) = p∀∀i (wkₚp (∈ₚ*▹tp s) pf)
|
||||
lliftₚ : {Γₚ : Conp Δₜ} → Δₚ ∈ₚ* Δₚ' → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} Δₚ' Γₚ
|
||||
lliftₚ s εₚ = εₚ
|
||||
lliftₚ s (σₚ ,ₚ pf) = lliftₚ s σₚ ,ₚ wkₚp s pf
|
||||
|
||||
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
|
||||
--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}
|
||||
|
||||
|
||||
-- We now can create Renamings, a subcategory from (Conp,Subp) that
|
||||
-- A renaming from a context Γₚ to a context Δₚ means that 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)
|
||||
data Ren : Conp Γₜ → Conp Γₜ → Set₁ where
|
||||
zeroRen : Ren ◇p Γₚ
|
||||
leftRen : {A : For Δₜ} → PfVar Δₜ Δₚ A → Ren Δₚ' Δₚ → Ren (Δₚ' ▹p⁰ A) Δₚ
|
||||
|
||||
-- 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
|
||||
|
||||
-- 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 → Set₁ where
|
||||
εₚ : Subp Δₚ ◇p
|
||||
_,ₚ_ : {A : For Δₜ} → (σ : Subp Δₚ Δₚ') → Pf Δₜ Δₚ A → Subp Δₚ (Δₚ' ▹p⁰ A)
|
||||
-- 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
|
||||
liftₚσ : {Δₜ : Cont}{Δₚ Γₚ : Conp Δₜ}{A : For Δₜ} → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} (Δₚ ▹p⁰ A) (Γₚ ▹p⁰ A)
|
||||
liftₚσ σ = (wkₚσt σ) ,ₚ (var pvzero)
|
||||
|
||||
idₚ : Subp {Δₜ} Δₚ Δₚ
|
||||
idₚ {Δₚ = ◇p} = εₚ
|
||||
idₚ {Δₚ = Δₚ ▹p⁰ x} = liftₚσ (idₚ {Δₚ = Δₚ})
|
||||
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)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
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-∘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)
|
||||
_[_]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 (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 [ lfₜσₜ 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))
|
||||
|
||||
_[_]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 = ◇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 (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
|
||||
_[_]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 [ lliftₚ (right∈ₚ* refl∈ₚ*) σ ,ₚ var pvzero ]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)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
idₚ : Subp {Δₜ} Δₚ Δₚ
|
||||
idₚ {Δₚ = ◇p} = εₚ
|
||||
idₚ {Δₚ = Δₚ ▹p⁰ x} = lfₚσₚ (idₚ {Δₚ = Δₚ})
|
||||
|
||||
_∘ₚ_ : {Γₚ Δₚ Ξₚ : Conp Δₜ} → Subp {Δₜ} Δₚ Ξₚ → Subp {Δₜ} Γₚ Δₚ → Subp {Δₜ} Γₚ Ξₚ
|
||||
εₚ ∘ₚ β = εₚ
|
||||
(α ,ₚ pf) ∘ₚ β = (α ∘ₚ β) ,ₚ (pf [ β ]p)
|
||||
∘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
|
||||
lemG' :
|
||||
{Γₜ Δₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Δₜ}{Ψₚ : Conp Δₜ}
|
||||
{αₜ : Subt Γₜ Δₜ}{γₚ : Subp Ξₚ Ψₚ}{βₚ : Subp Δₚ Ξₚ}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
|
||||
→ ((γₚ ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ ≡ (γₚ [ αₜ ]σₚ) ∘ₚ ((βₚ [ αₜ ]σₚ) ∘ₚ αₚ)
|
||||
lemG' {γₚ = εₚ} = refl
|
||||
lemG' {αₜ = αₜ}{γₚ ,ₚ x}{βₚ}{αₚ} = cong (λ ξ → ξ ,ₚ (((x [ βₚ ]p) [ αₜ ]pₜ) [ αₚ ]p)) (lemG' {γₚ = γₚ})
|
||||
ε-u : {Γₚ : Conp Γₜ} → {σ : Subp Γₚ ◇p} → σ ≡ εₚ {Δₚ = Γₚ}
|
||||
ε-u {σ = εₚ} = refl
|
||||
lemJ : {Δₜ : Cont}{Δₚ : Conp Δₜ}{A : For Δₜ} → Pf (con Δₜ Δₚ) A → (Pf (con Δₜ (Δₚ [ idₜ ]c)) (A [ idₜ ]f))
|
||||
lemJ {Δₜ}{Δₚ}{A} pf = substP (Pf (con Δₜ (Δₚ [ idₜ ]c))) (≡sym []f-id) (substP (λ Ξₚ → Pf (con Δₜ Ξₚ) A) (≡sym []c-id) pf)
|
||||
lemG :
|
||||
{Γₜ Δₜ Ξₜ Ψₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Ξₜ}{Ψₚ : Conp Ψₜ}
|
||||
{αₜ : Subt Γₜ Δₜ}{βₜ : Subt Δₜ Ξₜ}{γₜ : Subt Ξₜ Ψₜ}{γₚ : Subp Ξₚ (Ψₚ [ γₜ ]c)}{βₚ : Subp Δₚ (Ξₚ [ βₜ ]c)}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
|
||||
{eq₁ : Subp Γₚ (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c) ≡ Subp Γₚ (Ψₚ [ γₜ ∘ₜ (βₜ ∘ₜ αₜ) ]c)}
|
||||
{eq₂ : Subp (Δₚ [ αₜ ]c) ((Ψₚ [ γₜ ∘ₜ βₜ ]c) [ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c)}
|
||||
{eq₃ : Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ∘ₜ αₜ ]c) ≡ Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) (Ψₚ [ γₜ ∘ₜ (βₜ ∘ₜ αₜ) ]c)}
|
||||
{eq₄ : Subp (Δₚ [ αₜ ]c) ((Ξₚ [ βₜ ]c) [ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ξₚ [ βₜ ∘ₜ αₜ ]c)}
|
||||
{eq₅ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]c)}
|
||||
→ coe eq₁ ((coe eq₂ (((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ)) ∘ₚ αₚ) ≡ (coe eq₃ (γₚ [ βₜ ∘ₜ αₜ ]σₚ)) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ)
|
||||
lemK : {σₜ : Subt Γₜ Δₜ}{σₚ : Subp (Γₚ [ idₜ ]c) ((Δₚ [ σₜ ]c)[ idₜ ]c)}
|
||||
{eq1 : Subp Γₚ ((Δₚ [ σₜ ]c) [ idₜ ]c) ≡ Subp Γₚ (Δₚ [ σₜ ]c)}
|
||||
{eq2 : Subp Γₚ Γₚ ≡ Subp Γₚ (Γₚ [ idₜ ]c)}
|
||||
{eq3 : Subp (Γₚ [ idₜ ]c) ((Δₚ [ σₜ ]c)[ idₜ ]c) ≡ Subp Γₚ (Δₚ [ σₜ ]c)}
|
||||
→ coe eq1 (σₚ ∘ₚ coe eq2 idₚ)
|
||||
≡ (coe eq3 σₚ ∘ₚ idₚ)
|
||||
lemK {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})
|
||||
lemJ : {Δₜ : Cont}{Δₚ : Conp Δₜ}{A : For Δₜ} → Pf Δₜ Δₚ A → (Pf Δₜ (Δₚ [ idₜ ]c) (A [ idₜ ]f))
|
||||
lemJ {Δₜ}{Δₚ}{A} pf = substP (Pf Δₜ (Δₚ [ idₜ ]c)) (≡sym []f-id) (substP (λ Ξₚ → Pf Δₜ Ξₚ A) (≡sym []c-id) pf)
|
||||
[]σₚ-id : {σₚ : Subp {Δₜ} Δₚ Δₚ'} → coe (cong₂ Subp []c-id []c-id) (σₚ [ idₜ ]σₚ) ≡ σₚ
|
||||
[]σₚ-id {σₚ = εₚ} = ε-u
|
||||
[]σₚ-id {Δₜ}{Δₚ}{Δₚ' ▹p⁰ A} {σₚ = σₚ ,ₚ x} = substP (λ ξ → coe (cong₂ Subp []c-id []c-id) (ξ ,ₚ (x [ idₜ ]pₜ)) ≡ (σₚ ,ₚ x)) (≡sym (coeshift ([]σₚ-id)))
|
||||
(≡sym (coeshift {eq = cong₂ Subp (≡sym []c-id) (≡sym []c-id)}
|
||||
(substfpoly'' {A = (Conp Δₜ) × (Conp Δₜ)}{P = λ W F → Subp (proj×₁ W) ((proj×₂ W) ▹p⁰ F)}{R = λ W → Subp (proj×₁ W) (proj×₂ W)}{Q = λ W F → Pf (con Δₜ (proj×₁ W)) F}{α = Δₚ ,× Δₚ'}{ε = A}{eq = ×≡ (≡sym []c-id) (≡sym []c-id)}{eq'' = ≡sym []f-id}{f = λ {W} {F} ξ pf → _,ₚ_ ξ pf}{x = σₚ}{y = x})))
|
||||
(substfpoly'' {A = (Conp Δₜ) × (Conp Δₜ)}{P = λ W F → Subp (proj×₁ W) ((proj×₂ W) ▹p⁰ F)}{R = λ W → Subp (proj×₁ W) (proj×₂ W)}{Q = λ W F → Pf Δₜ (proj×₁ W) F}{α = Δₚ ,× Δₚ'}{ε = A}{eq = ×≡ (≡sym []c-id) (≡sym []c-id)}{eq'' = ≡sym []f-id}{f = λ {W} {F} ξ pf → _,ₚ_ ξ pf}{x = σₚ}{y = x})))
|
||||
[]σₚ-∘ : {Ξₚ Ξₚ' : Conp Ξₜ}{α : Subt Δₜ Ξₜ} {β : Subt Γₜ Δₜ} {σₚ : Subp {Ξₜ} Ξₚ Ξₚ'}
|
||||
--{eq₅ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]c)}
|
||||
→ coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) ((σₚ [ α ]σₚ) [ β ]σₚ) ≡ σₚ [ α ∘ₜ β ]σₚ
|
||||
@ -362,62 +379,11 @@ module FFOLInitial where
|
||||
{A = (Conp Γₜ) × (Conp Γₜ)}
|
||||
{P = λ W F → Subp (proj×₁ W) ((proj×₂ W) ▹p⁰ F)}
|
||||
{R = λ W → Subp (proj×₁ W) (proj×₂ W)}
|
||||
{Q = λ W F → Pf (con Γₜ (proj×₁ W)) F}
|
||||
{Q = λ W F → Pf Γₜ (proj×₁ W) F}
|
||||
{eq = cong₂ _,×_ []c-∘ []c-∘}
|
||||
{eq'' = []f-∘ {α = β} {β = α} {F = A}}
|
||||
{f = λ {W} {F} ξ pf → _,ₚ_ ξ pf}{x = σₚ [ α ∘ₜ β ]σₚ}{y = pf [ α ∘ₜ β ]pₜ})
|
||||
))
|
||||
wkₚ∘, : {Δₜ : Cont}{Γₚ Δₚ Ξₚ : Conp Δₜ}{α : Subp {Δₜ} Γₚ Δₚ}{β : Subp {Δₜ} Ξₚ Γₚ}{A : For Δₜ}{pf : Pf (con Δₜ Ξₚ) A} → (wkₚσt α) ∘ₚ (β ,ₚ pf) ≡ (α ∘ₚ β)
|
||||
wkₚ∘, {α = εₚ} = refl
|
||||
wkₚ∘, {α = α ,ₚ pf} {β = β} {pf = pf'} = cong (λ ξ → ξ ,ₚ (pf [ β ]p)) wkₚ∘,
|
||||
idlₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → (idₚ {Δₚ = Δₚ}) ∘ₚ σₚ ≡ σₚ
|
||||
idlₚ {Δₚ = ◇p} {εₚ} = refl
|
||||
idlₚ {Δₚ = Δₚ ▹p⁰ pf} {σₚ ,ₚ pf'} = cong (λ ξ → ξ ,ₚ pf') (≡tran wkₚ∘, (idlₚ {σₚ = σₚ}))
|
||||
idrₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → σₚ ∘ₚ (idₚ {Δₚ = Γₚ}) ≡ σₚ
|
||||
idrₚ {σₚ = εₚ} = refl
|
||||
idrₚ {σₚ = σₚ ,ₚ prf} = cong (λ X → X ,ₚ prf) (idrₚ {σₚ = σₚ})
|
||||
wkₚ[] : {σₜ : Subt Γₜ Δₜ} {σₚ : Subp Δₚ Δₚ'} {A : For Δₜ} → (wkₚσt {A = A} σₚ) [ σₜ ]σₚ ≡ wkₚσt (σₚ [ σₜ ]σₚ)
|
||||
wkₚ[] {σₚ = εₚ} = refl
|
||||
wkₚ[] {σₚ = σₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ _) (wkₚ[] {σₚ = σₚ})
|
||||
idₚ[] : {σₜ : Subt Γₜ Δₜ} → ((idₚ {Δₜ} {Δₚ}) [ σₜ ]σₚ) ≡ idₚ {Γₜ} {Δₚ [ σₜ ]c}
|
||||
idₚ[] {Δₚ = ◇p} = refl
|
||||
idₚ[] {Δₚ = Δₚ ▹p⁰ A} = cong (λ ξ → ξ ,ₚ var pvzero) (≡tran wkₚ[] (cong wkₚσt idₚ[]))
|
||||
|
||||
|
||||
id : Sub Γ Γ
|
||||
id {Γ} = sub idₜ (subst (Subp _) (≡sym []c-id) idₚ)
|
||||
_∘_ : 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 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)}
|
||||
{eq3 : Subp (Con.p Γ [ idₜ ]c) ((Con.p Δ [ σₜ ]c)[ idₜ ]c) ≡ Subp (Con.p Γ) (Con.p Δ [ σₜ ]c)}
|
||||
→ coe eq1 (σₚ ∘ₚ coe eq2 idₚ)
|
||||
≡ (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 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
|
||||
|
||||
lemG' :
|
||||
{Γₜ Δₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Δₜ}{Ψₚ : Conp Δₜ}
|
||||
{αₜ : Subt Γₜ Δₜ}{γₚ : Subp Ξₚ Ψₚ}{βₚ : Subp Δₚ Ξₚ}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
|
||||
→ ((γₚ ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ ≡ (γₚ [ αₜ ]σₚ) ∘ₚ ((βₚ [ αₜ ]σₚ) ∘ₚ αₚ)
|
||||
lemG' {γₚ = εₚ} = refl
|
||||
lemG' {αₜ = αₜ}{γₚ ,ₚ x}{βₚ}{αₚ} = cong (λ ξ → ξ ,ₚ (((x [ βₚ ]p) [ αₜ ]pₜ) [ αₚ ]p)) (lemG' {γₚ = γₚ})
|
||||
lemG :
|
||||
{Γₜ Δₜ Ξₜ Ψₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Ξₜ}{Ψₚ : Conp Ψₜ}
|
||||
{αₜ : Subt Γₜ Δₜ}{βₜ : Subt Δₜ Ξₜ}{γₜ : Subt Ξₜ Ψₜ}{γₚ : Subp Ξₚ (Ψₚ [ γₜ ]c)}{βₚ : Subp Δₚ (Ξₚ [ βₜ ]c)}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
|
||||
{eq₁ : Subp Γₚ (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c) ≡ Subp Γₚ (Ψₚ [ γₜ ∘ₜ (βₜ ∘ₜ αₜ) ]c)}
|
||||
{eq₂ : Subp (Δₚ [ αₜ ]c) ((Ψₚ [ γₜ ∘ₜ βₜ ]c) [ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c)}
|
||||
{eq₃ : Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ∘ₜ αₜ ]c) ≡ Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) (Ψₚ [ γₜ ∘ₜ (βₜ ∘ₜ αₜ) ]c)}
|
||||
{eq₄ : Subp (Δₚ [ αₜ ]c) ((Ξₚ [ βₜ ]c) [ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ξₚ [ βₜ ∘ₜ αₜ ]c)}
|
||||
{eq₅ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]c)}
|
||||
→ coe eq₁ ((coe eq₂ (((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ)) ∘ₚ αₚ) ≡ (coe eq₃ (γₚ [ βₜ ∘ₜ αₜ ]σₚ)) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ)
|
||||
lemG {Γₜ}{Δₜ}{Ξₜ}{Ψₜ}{Γₚ}{Δₚ}{Ξₚ}{Ψₚ}{αₜ = αₜ}{βₜ}{γₜ}{γₚ}{βₚ}{αₚ}{eq₁}{eq₂}{eq₃}{eq₄}{eq₅} =
|
||||
substP (λ X → coe eq₁ ((coe eq₂ (((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ)) ∘ₚ αₚ) ≡ (coe eq₃ X) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ)) []σₚ-∘ (
|
||||
≡tran⁵
|
||||
@ -452,20 +418,63 @@ module FFOLInitial where
|
||||
{f = λ {τ} ξ → (ξ ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ))}
|
||||
{x = (coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) ((γₚ [ βₜ ]σₚ) [ αₜ ]σₚ))}
|
||||
))
|
||||
wkₚ∘, : {Δₜ : Cont}{Γₚ Δₚ Ξₚ : Conp Δₜ}{α : Subp {Δₜ} Γₚ Δₚ}{β : Subp {Δₜ} Ξₚ Γₚ}{A : For Δₜ}{pf : Pf Δₜ Ξₚ A} → (wkₚσₚ α) ∘ₚ (β ,ₚ pf) ≡ (α ∘ₚ β)
|
||||
wkₚ∘, {α = εₚ} = refl
|
||||
wkₚ∘, {α = α ,ₚ pf} {β = β} {pf = pf'} = cong (λ ξ → ξ ,ₚ (pf [ β ]p)) wkₚ∘,
|
||||
idlₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → (idₚ {Δₚ = Δₚ}) ∘ₚ σₚ ≡ σₚ
|
||||
idlₚ {Δₚ = ◇p} {εₚ} = refl
|
||||
idlₚ {Δₚ = Δₚ ▹p⁰ pf} {σₚ ,ₚ pf'} = cong (λ ξ → ξ ,ₚ pf') (≡tran wkₚ∘, (idlₚ {σₚ = σₚ}))
|
||||
idrₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → σₚ ∘ₚ (idₚ {Δₚ = Γₚ}) ≡ σₚ
|
||||
idrₚ {σₚ = εₚ} = refl
|
||||
idrₚ {σₚ = σₚ ,ₚ prf} = cong (λ X → X ,ₚ prf) (idrₚ {σₚ = σₚ})
|
||||
wkₚ[] : {σₜ : Subt Γₜ Δₜ} {σₚ : Subp Δₚ Δₚ'} {A : For Δₜ} → (wkₚσₚ {A = A} σₚ) [ σₜ ]σₚ ≡ wkₚσₚ (σₚ [ σₜ ]σₚ)
|
||||
wkₚ[] {σₚ = εₚ} = refl
|
||||
wkₚ[] {σₚ = σₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ _) (wkₚ[] {σₚ = σₚ})
|
||||
idₚ[] : {σₜ : Subt Γₜ Δₜ} → ((idₚ {Δₜ} {Δₚ}) [ σₜ ]σₚ) ≡ idₚ {Γₜ} {Δₚ [ σₜ ]c}
|
||||
idₚ[] {Δₚ = ◇p} = refl
|
||||
idₚ[] {Δₚ = Δₚ ▹p⁰ A} = cong (λ ξ → ξ ,ₚ var pvzero) (≡tran wkₚ[] (cong wkₚσₚ idₚ[]))
|
||||
|
||||
|
||||
record Con : Set₁ where
|
||||
constructor con
|
||||
field
|
||||
t : Cont
|
||||
p : Conp t
|
||||
|
||||
◇ : Con
|
||||
◇ = con ◇t ◇p
|
||||
_▹p_ : (Γ : Con) → For (Con.t Γ) → Con
|
||||
Γ ▹p A = con (Con.t Γ) (Con.p Γ ▹p⁰ A)
|
||||
|
||||
variable
|
||||
Γ Δ Ξ : Con
|
||||
_▹t : Con → Con
|
||||
Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp)
|
||||
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)
|
||||
id : Sub Γ Γ
|
||||
id {Γ} = sub idₜ (subst (Subp _) (≡sym []c-id) idₚ)
|
||||
_∘_ : 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 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))
|
||||
idr : {Γ Δ : Con} {σ : Sub Γ Δ} → σ ∘ (id {Γ}) ≡ σ
|
||||
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)
|
||||
|
||||
∘-ass : {Γ Δ Ξ Ψ : Con}{α : Sub Γ Δ}{β : Sub Δ Ξ}{γ : Sub Ξ Ψ} → (γ ∘ β) ∘ α ≡ γ ∘ (β ∘ α)
|
||||
∘-ass {Γ}{Δ}{Ξ}{Ψ}{α = sub αₜ αₚ} {β = sub βₜ βₚ} {γ = sub γₜ γₚ} = cong₂' sub ∘ₜ-ass lemG
|
||||
|
||||
-- SUB-ization
|
||||
|
||||
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 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)
|
||||
πₜ¹* (sub (σₜ ,ₜ t) σₚ) = sub σₜ (subst (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) (subst (Subp _) (≡sym lemA) σₚ)
|
||||
(sub σₜ σₚ) ,ₜ* t = sub (σₜ ,ₜ t) (subst (Subp _) (≡sym ▹tp,ₜ) σₚ)
|
||||
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
|
||||
@ -473,55 +482,35 @@ module FFOLInitial where
|
||||
,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} = cong (sub (σₜ ,ₜ t)) coeaba
|
||||
,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)
|
||||
lemE : {σₜ : Subt Γₜ Ξₜ}{σₚ : Subp Γₚ (Ξₚ [ σₜ ]c)} {δₜ : Subt Δₜ Γₜ} → (coe _ σₚ [ δₜ ]σₚ) ≡ coe _ (σₚ [ δₜ ]σₚ)
|
||||
lemE {δₜ = δₜ} = coecong {eq = refl} {eq' = refl} (λ ξₚ → ξₚ [ δₜ ]σₚ)
|
||||
lemF : {Γα Γβ : Conp Δₜ}{δₜ : Subt Δₜ Γₜ}{δₚ : Subp Δₚ (Γₚ [ δₜ ]c)} → (eq : Γβ ≡ Γα) → (ξ : Subp (Γₚ [ δₜ ]c) Γβ) → coe (cong (Subp Δₚ) eq) (ξ ∘ₚ δₚ) ≡ coe (cong (Subp _) eq) ξ ∘ₚ δₚ
|
||||
lemF refl ξ = ≡tran coerefl (cong₂ _∘ₚ_ (≡sym coerefl) refl)
|
||||
--lemG : {Γα Γβ : Conp Δₜ}{σₜ : Subt Γₜ Ξₜ}{δₜ : Subt Δₜ Γₜ} → (eq : Γβ ≡ Γα) → (ξ : Subp Γₚ (Ξₚ [ σₜ ]c)) → coe refl (ξ [ δₜ ]σₚ) ≡ (coe refl ξ) [ δₜ ]σₚ
|
||||
--lemG eq ε= {!!}
|
||||
substf : {ℓ ℓ' : Level}{A : Set ℓ}{P : A → Set ℓ'}{Q : A → Set ℓ'}{a b c d : A}{eqa : a ≡ a}{eqb : b ≡ b}{eqcd : c ≡ d}{eqdc : d ≡ c}{f : P a → P b}{g : P b → Q c}{x : P a} → g (subst P eqb (f (subst P eqa x))) ≡ subst Q eqdc (subst Q eqcd (g (f x)))
|
||||
substf {P = P} {Q = Q} {eqcd = refl} {f = f} {g = g} = ≡tran² (cong g (≡tran (substrefl {P = P} {e = refl}) (cong f (substrefl {P = P} {e = refl})))) (≡sym (substrefl {P = Q} {e = refl})) (≡sym (substrefl {P = Q} {e = refl}))
|
||||
|
||||
,ₜ∘* {Γ} {Δ} {Ξ} {sub σₜ σₚ} {sub δₜ δₚ} {t} = cong (sub ((σₜ ∘ₜ δₜ) ,ₜ (t [ δₜ ]t)))
|
||||
(substfgpoly
|
||||
{P = Subp {Con.t Δ} (Con.p Δ)}
|
||||
{Q = Subp {Con.t Δ} ((Con.p Γ) [ δₜ ]c)}
|
||||
{R = Subp {Con.t Γ} (Con.p Γ)}
|
||||
{F = λ X → X [ δₜ ]c}
|
||||
{eq₁ = ≡sym lemA}
|
||||
{eq₁ = ≡sym ▹tp,ₜ}
|
||||
{eq₂ = ≡sym []c-∘}
|
||||
{eq₃ = ≡sym []c-∘}
|
||||
{eq₄ = ≡sym lemA}
|
||||
{eq₄ = ≡sym ▹tp,ₜ}
|
||||
{g = λ σₚ → σₚ ∘ₚ δₚ}
|
||||
{f = λ σₚ → σₚ [ δₜ ]σₚ}
|
||||
{x = σₚ})
|
||||
|
||||
πₚ¹* : {Γ Δ : Con} {A : For (Con.t Γ)} → Sub Δ (Γ ▹p A) → Sub Δ Γ
|
||||
πₚ¹* (sub σₜ (σₚ ,ₚ pf)) = sub σₜ σₚ
|
||||
πₚ² : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ (Γ ▹p F)) → Pf Δ (F [ Sub.t (πₚ¹* σ) ]f)
|
||||
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 Δ (F [ Sub.t σ ]f) → Sub Δ (Γ ▹p F)
|
||||
_,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)
|
||||
|
||||
,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 Γ (F [ Sub.t σ ]f)}
|
||||
→ (σ ,ₚ* prf) ∘ δ ≡ (σ ∘ δ) ,ₚ* (substP (λ F → Pf Δ F) (≡sym []f-∘) ((prf [ Sub.t δ ]pₜ) [ Sub.p δ ]p))
|
||||
,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} = cong (sub (σₜ ∘ₜ δₜ)) (cong (λ ξ → ξ ∘ₚ δₚ)
|
||||
(substfpoly⁴ {P = λ W → Subp (Con.p Γ [ δₜ ]c) ((proj×₁ W) ▹p⁰ (proj×₂ W))}{R = λ W → Subp (Con.p Γ [ δₜ ]c) (proj×₁ W)}{Q = λ W → Pf (con (Con.t Δ) (Con.p Γ [ δₜ ]c)) (proj×₂ W)}{α = ((Con.p Ξ [ σₜ ]c) [ δₜ ]c) ,× ((A [ σₜ ]f) [ δₜ ]f)}{eq = cong₂ _,×_ (≡sym []c-∘) (≡sym []f-∘)}{f = λ ξ p → ξ ,ₚ p} {x = σₚ [ δₜ ]σₚ}{y = prf [ δₜ ]pₜ})) --
|
||||
(substfpoly⁴ {P = λ W → Subp (Con.p Γ [ δₜ ]c) ((proj×₁ W) ▹p⁰ (proj×₂ W))}{R = λ W → Subp (Con.p Γ [ δₜ ]c) (proj×₁ W)}{Q = λ W → Pf (Con.t Δ) (Con.p Γ [ δₜ ]c) (proj×₂ W)}{α = ((Con.p Ξ [ σₜ ]c) [ δₜ ]c) ,× ((A [ σₜ ]f) [ δₜ ]f)}{eq = cong₂ _,×_ (≡sym []c-∘) (≡sym []f-∘)}{f = λ ξ p → ξ ,ₚ p} {x = σₚ [ δₜ ]σₚ}{y = prf [ δₜ ]pₜ})) --
|
||||
|
||||
--_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹t)
|
||||
--πₜ²∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ² (σ ,ₜ t) ≡ t
|
||||
--πₜ¹∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ¹ (σ ,ₜ t) ≡ σ
|
||||
--,ₜ∘πₜ : {Γ Δ : Con} → {σ : Sub Δ (Γ ▹ₜ)} → (πₜ¹ σ) ,ₜ (πₜ² σ) ≡ σ
|
||||
--,ₜ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{t : Tm Γ} → (σ ,ₜ t) ∘ δ ≡ (σ ∘ δ) ,ₜ (t [ δ ]t)
|
||||
|
||||
-- lemB : ∀{ℓ}{A : Set ℓ}{ℓ'}{P : A → Set ℓ'}{a a' : A}{e : a ≡ a'}{p : P a}{p' : P a'} → p' ≡ p → subst P e p' ≡ p
|
||||
|
||||
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 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
|
||||
|
||||
@ -556,7 +545,7 @@ module FFOLInitial where
|
||||
; []f-∘ = []f-∘
|
||||
; R = R
|
||||
; R[] = refl
|
||||
; _⊢_ = Pf
|
||||
; _⊢_ = 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*
|
||||
@ -569,7 +558,7 @@ module FFOLInitial where
|
||||
; []f-⇒ = refl
|
||||
; ∀∀ = ∀∀
|
||||
; []f-∀∀ = []f-∀∀
|
||||
; lam = λ {Γ}{F}{G} pf → substP (λ H → Pf Γ (F ⇒ H)) (≡tran (cong (_[_]f G) (lemD {σ = id})) []f-id) (lam pf)
|
||||
; lam = λ {Γ}{F}{G} pf → substP (λ H → Pf (Con.t Γ) (Con.p Γ) (F ⇒ H)) (≡tran (cong (_[_]f G) (lemD {σ = id})) []f-id) (lam pf)
|
||||
; app = app
|
||||
; ∀i = p∀∀i
|
||||
; ∀e = λ {Γ} {F} pf {t} → p∀∀e pf
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user