From 824a10d5d227e924ea3d2b68438f13d4444339d7 Mon Sep 17 00:00:00 2001 From: Mysaa Date: Wed, 19 Jul 2023 16:49:48 +0200 Subject: [PATCH] I FINALLY HAVE A SYNTAX !!!! --- FFOLCompleteness.agda | 327 ++++++++++++++++++++++++++++++++++++++++++ FFOLInitial.agda | 163 ++++++++++++++++----- PropUtil.agda | 98 ++++++++++++- 3 files changed, 549 insertions(+), 39 deletions(-) create mode 100644 FFOLCompleteness.agda diff --git a/FFOLCompleteness.agda b/FFOLCompleteness.agda new file mode 100644 index 0000000..1c5b8e8 --- /dev/null +++ b/FFOLCompleteness.agda @@ -0,0 +1,327 @@ +{-# OPTIONS --prop --rewriting #-} + +open import PropUtil + +module FFOLCompleteness where + + open import Agda.Primitive + open import FinitaryFirstOrderLogic + open import ListUtil + + record Family : Set (lsuc (ℓ¹)) where + field + World : Set ℓ¹ + _≤_ : World → World → Prop + ≤refl : {w : World} → w ≤ w + ≤tran : {w w' w'' : World} → w ≤ w' → w' ≤ w'' → w ≤ w' + TM : World → Set ℓ¹ + TM≤ : {w w' : World} → w ≤ w' → TM w → TM w' + REL : (w : World) → TM w → TM w → Prop ℓ¹ + REL≤ : {w w' : World} → {t u : TM w} → (eq : w ≤ w') → REL w t u → REL w' (TM≤ eq t) (TM≤ eq u) + infixr 10 _∘_ + Con = World → Set ℓ¹ + Sub : Con → Con → Set ℓ¹ + Sub Δ Γ = (w : World) → Δ w → Γ w + _∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ + α ∘ β = λ w γ → α w (β w γ) + id : {Γ : Con} → Sub Γ Γ + id = λ w γ → γ + ◇ : Con -- The initial object of the category + ◇ = λ w → ⊤ₛ + ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object + ε w Γ = ttₛ + + -- Functor Con → Set called Tm + Tm : Con → Set ℓ¹ + Tm Γ = (w : World) → (Γ w) → TM w + _[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms + t [ σ ]t = λ w → λ γ → t w (σ w γ) + + -- Tm⁺ + _▹ₜ : Con → Con + Γ ▹ₜ = λ w → (Γ w) × (TM w) + πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ + πₜ¹ σ = λ w → λ x → proj×₁ (σ w x) + πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ + πₜ² σ = λ w → λ x → proj×₂ (σ w x) + _,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ) + σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x) + + -- Functor Con → Set called For + For : Con → Set (lsuc ℓ¹) + For Γ = (w : World) → (Γ w) → Prop ℓ¹ + _[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms + F [ σ ]f = λ w → λ x → F w (σ w x) + + -- Formulas with relation on terms + R : {Γ : Con} → Tm Γ → Tm Γ → For Γ + R t u = λ w → λ γ → REL w (t w γ) (u w γ) + + + -- Proofs + _⊢_ : (Γ : Con) → For Γ → Prop ℓ¹ + Γ ⊢ F = ∀ w (γ : Γ w) → F w γ + _[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms + prf [ σ ]p = λ w → λ γ → prf w (σ w γ) + -- Equalities below are useless because Γ ⊢ F is in prop + -- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf + -- []p-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ α ∘ β ]p ≡ (prf [ β ]p) [ α ]p + + -- → Prop⁺ + _▹ₚ_ : (Γ : Con) → For Γ → Con + Γ ▹ₚ F = λ w → (Γ w) ×'' (F w) + πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ + πₚ¹ σ w δ = proj×''₁ (σ w δ) + πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f) + πₚ² σ w δ = proj×''₂ (σ w δ) + _,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F) + _,ₚ_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ + + + + -- Implication + _⇒_ : {Γ : Con} → For Γ → For Γ → For Γ + F ⇒ G = λ w → λ γ → (∀ w' → w ≤ w' → (F w γ) → (G w γ)) + + -- Forall + ∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ + ∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t) + + -- Lam & App + lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G) + lam prf = λ w γ w' s h → prf w (γ ,×'' h) + app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G + app prf prf' = λ w γ → prf w γ w ≤refl (prf' w γ) + -- 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) + ∀i p w γ = λ t → p w (γ ,× t) + ∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f) + ∀e p {t} w γ = p w γ (t w γ) + + + tod : FFOL + tod = record + { Con = Con + ; Sub = Sub + ; _∘_ = _∘_ + ; ∘-ass = refl + ; id = id + ; idl = refl + ; idr = refl + ; ◇ = ◇ + ; ε = ε + ; ε-u = refl + ; Tm = Tm + ; _[_]t = _[_]t + ; []t-id = refl + ; []t-∘ = refl + ; _▹ₜ = _▹ₜ + ; πₜ¹ = πₜ¹ + ; πₜ² = πₜ² + ; _,ₜ_ = _,ₜ_ + ; πₜ²∘,ₜ = refl + ; πₜ¹∘,ₜ = refl + ; ,ₜ∘πₜ = refl + ; ,ₜ∘ = refl + ; For = For + ; _[_]f = _[_]f + ; []f-id = refl + ; []f-∘ = refl + ; _⊢_ = _⊢_ + ; _[_]p = _[_]p + ; _▹ₚ_ = _▹ₚ_ + ; πₚ¹ = πₚ¹ + ; πₚ² = πₚ² + ; _,ₚ_ = _,ₚ_ + ; ,ₚ∘πₚ = refl + ; πₚ¹∘,ₚ = refl + ; ,ₚ∘ = refl + ; _⇒_ = _⇒_ + ; []f-⇒ = refl + ; ∀∀ = ∀∀ + ; []f-∀∀ = refl + ; lam = lam + ; app = app + ; ∀i = ∀i + ; ∀e = ∀e + ; R = R + ; R[] = refl + } + + record Presheaf : Set (lsuc (ℓ¹)) where + field + World : Set ℓ¹ + _≤_ : World → World → Set ℓ¹ -- arrows + ≤refl : {w : World} → w ≤ w -- id arrow + ≤tran : {w w' w'' : World} → w ≤ w' → w' ≤ w'' → w ≤ w'' -- arrow composition + ≤-ass : {w w' w'' w''' : World}{a : w ≤ w'}{b : w' ≤ w''}{c : w'' ≤ w'''} + → (≤tran (≤tran a b) c) ≡ (≤tran a (≤tran b c)) + ≤-idl : {w w' : World} → {a : w ≤ w'} → ≤tran (≤refl {w}) a ≡ a + ≤-idr : {w w' : World} → {a : w ≤ w'} → ≤tran a (≤refl {w'}) ≡ a + TM : World → Set ℓ¹ + TM≤ : {w w' : World} → w ≤ w' → TM w → TM w' + REL : (w : World) → TM w → TM w → Prop ℓ¹ + REL≤ : {w w' : World} → {t u : TM w} → (eq : w ≤ w') → REL w t u → REL w' (TM≤ eq t) (TM≤ eq u) + infixr 10 _∘_ + Con = World → Set ℓ¹ + Sub : Con → Con → Set ℓ¹ + Sub Δ Γ = (w : World) → Δ w → Γ w + _∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ + α ∘ β = λ w γ → α w (β w γ) + id : {Γ : Con} → Sub Γ Γ + id = λ w γ → γ + ◇ : Con -- The initial object of the category + ◇ = λ w → ⊤ₛ + ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object + ε w Γ = ttₛ + + -- Functor Con → Set called Tm + Tm : Con → Set ℓ¹ + Tm Γ = (w : World) → (Γ w) → TM w + _[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms + t [ σ ]t = λ w → λ γ → t w (σ w γ) + + -- Tm⁺ + _▹ₜ : Con → Con + Γ ▹ₜ = λ w → (Γ w) × (TM w) + πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ + πₜ¹ σ = λ w → λ x → proj×₁ (σ w x) + πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ + πₜ² σ = λ w → λ x → proj×₂ (σ w x) + _,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ) + σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x) + + -- Functor Con → Set called For + For : Con → Set (lsuc ℓ¹) + For Γ = (w : World) → (Γ w) → Prop ℓ¹ + _[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms + F [ σ ]f = λ w → λ x → F w (σ w x) + + -- Formulas with relation on terms + R : {Γ : Con} → Tm Γ → Tm Γ → For Γ + R t u = λ w → λ γ → REL w (t w γ) (u w γ) + + + -- Proofs + _⊢_ : (Γ : Con) → For Γ → Prop ℓ¹ + Γ ⊢ F = ∀ w (γ : Γ w) → F w γ + _[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms + prf [ σ ]p = λ w → λ γ → prf w (σ w γ) + -- Equalities below are useless because Γ ⊢ F is in prop + -- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf + -- []p-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ α ∘ β ]p ≡ (prf [ β ]p) [ α ]p + + -- → Prop⁺ + _▹ₚ_ : (Γ : Con) → For Γ → Con + Γ ▹ₚ F = λ w → (Γ w) ×'' (F w) + πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ + πₚ¹ σ w δ = proj×''₁ (σ w δ) + πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f) + πₚ² σ w δ = proj×''₂ (σ w δ) + _,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F) + _,ₚ_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ + + + + -- Implication + _⇒_ : {Γ : Con} → For Γ → For Γ → For Γ + F ⇒ G = λ w → λ γ → (∀ w' → w ≤ w' → (F w γ) → (G w γ)) + + -- Forall + ∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ + ∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t) + + -- Lam & App + lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G) + lam prf = λ w γ w' s h → prf w (γ ,×'' h) + app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G + app prf prf' = λ w γ → prf w γ w ≤refl (prf' w γ) + -- 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) + ∀i p w γ = λ t → p w (γ ,× t) + ∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f) + ∀e p {t} w γ = p w γ (t w γ) + + + tod : FFOL + tod = record + { Con = Con + ; Sub = Sub + ; _∘_ = _∘_ + ; ∘-ass = refl + ; id = id + ; idl = refl + ; idr = refl + ; ◇ = ◇ + ; ε = ε + ; ε-u = refl + ; Tm = Tm + ; _[_]t = _[_]t + ; []t-id = refl + ; []t-∘ = refl + ; _▹ₜ = _▹ₜ + ; πₜ¹ = πₜ¹ + ; πₜ² = πₜ² + ; _,ₜ_ = _,ₜ_ + ; πₜ²∘,ₜ = refl + ; πₜ¹∘,ₜ = refl + ; ,ₜ∘πₜ = refl + ; ,ₜ∘ = refl + ; For = For + ; _[_]f = _[_]f + ; []f-id = refl + ; []f-∘ = refl + ; _⊢_ = _⊢_ + ; _[_]p = _[_]p + ; _▹ₚ_ = _▹ₚ_ + ; πₚ¹ = πₚ¹ + ; πₚ² = πₚ² + ; _,ₚ_ = _,ₚ_ + ; ,ₚ∘πₚ = refl + ; πₚ¹∘,ₚ = refl + ; ,ₚ∘ = refl + ; _⇒_ = _⇒_ + ; []f-⇒ = refl + ; ∀∀ = ∀∀ + ; []f-∀∀ = refl + ; lam = lam + ; app = app + ; ∀i = ∀i + ; ∀e = ∀e + ; R = R + ; R[] = refl + } + + -- Completeness proof + + -- We first build our universal Kripke model + + module ComplenessProof where + + -- We have a model, we construct the Universal Presheaf model of this model + open import FFOLInitial as I + + World : Set₁ + World = I.Con + + _≤_ : World → World → Set₁ + Γ ≤ Δ = I.Sub Γ Δ + + UP : Presheaf + UP = record + { World = I.Con + ; _≤_ = I.Sub + ; ≤refl = I.id + ; ≤tran = λ σ σ' → σ' I.∘ σ + ; ≤-ass = λ {w}{w'}{w''}{w'''}{a}{b}{c} → ≡sym I.∘-ass + ; ≤-idl = I.idr + ; ≤-idr = I.idl + ; TM = λ Γ → I.Tm (Con.t Γ) + ; TM≤ = {!!} + ; REL = λ Γ t u → {!I.r t u!} + ; REL≤ = {!!} + } diff --git a/FFOLInitial.agda b/FFOLInitial.agda index 53a2974..2bf0162 100644 --- a/FFOLInitial.agda +++ b/FFOLInitial.agda @@ -134,7 +134,8 @@ module FFOLInitial where ∘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)) - + εₜ-u : {σₜ : Subt Γₜ ◇t} → σₜ ≡ εₜ + εₜ-u {σₜ = εₜ} = refl data Conp : Cont → Set₁ -- pu tit in Prop variable @@ -240,8 +241,6 @@ module FFOLInitial where ∈ₚ*→Sub zero∈ₚ* = εₚ ∈ₚ*→Sub (next∈ₚ* x s) = ∈ₚ*→Sub s ,ₚ var x - idₚ : Subp {Δₜ} Δₚ Δₚ - idₚ = ∈ₚ*→Sub refl∈ₚ* 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) @@ -253,8 +252,23 @@ module FFOLInitial where 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 + --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 + 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}) + @@ -282,7 +296,6 @@ module FFOLInitial where 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 @@ -302,33 +315,118 @@ module FFOLInitial where _∘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) + 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) + []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}))) + []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 + []σₚ-∘ {σₚ = εₚ} = ε-u + []σₚ-∘ {Ξₜ}{Δₜ}{Γₜ}{Ξₚ}{Δₚ' ▹p⁰ A}{α}{β}{σₚ = σₚ ,ₚ pf} = + substP (λ ξ → coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) (ξ ,ₚ ((pf [ α ]pₜ) [ β ]pₜ)) ≡ ((σₚ [ α ∘ₜ β ]σₚ) ,ₚ (pf [ α ∘ₜ β ]pₜ))) (≡sym (coeshift []σₚ-∘)) + (≡sym (coeshift {eq = cong₂ Subp []c-∘ []c-∘} + (substfpoly'' + {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} + {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} = ? - idlₚ {Δₚ = Δₚ ▹p⁰ x} = ? + 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 = {!!} + 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 {σ = sub σₜ σₚ} = cong₂' sub σ-idl {!!} + 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)) + 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 {σ = sub σₜ σₚ} = cong₂' sub σ-idr {!!} - {- - ∘ₚ-ass : + 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) + ∘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 (Δₚ [ αₜ ]c) ((Ψₚ [ γₜ ∘ₜ βₜ ]c)[ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c)} - {eq₂ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c)[ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]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)!}} + {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)} - → (coe eq₁ (((coe eq₂ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ) ≡ (coe eq₃ (γₚ [ βₜ ∘ₜ αₜ ]σₚ)) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ) ∘ₚ αₚ)) - -} - postulate ∘-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} = {!Subp (Con.p Ξ [ βₜ ∘ₜ αₜ ]c) (Con.p Ψ [ γₜ ∘ₜ (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 + (cong (coe eq₁) (≡tran ( + ≡sym (substfpoly + {R = λ X → Subp (Δₚ [ αₜ ]c) X} + {eq = ≡sym []c-∘} + {f = λ ξ → ξ ∘ₚ αₚ} + {x = ((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ})) + (cong (coe (cong (λ z → Subp Γₚ z) (≡sym []c-∘))) + (≡sym (substfpoly + {R = λ X → Subp (Ξₚ [ βₜ ]c) X} + {eq = ≡sym []c-∘} + {f = λ ξ → ((ξ ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ} + {x = γₚ [ βₜ ]σₚ} + ))) + )) + (≡tran coecoe-coe coecoe-coe) + (cong (coe (≡tran (cong (λ z → Subp Γₚ (z [ αₜ ]c)) (≡sym []c-∘)) (≡tran (cong (λ z → Subp Γₚ z) (≡sym []c-∘)) eq₁))) lemG') + (≡sym coecoe-coe) + (cong (coe (cong (λ z → Subp Γₚ z) (≡sym []c-∘))) (substppoly + {A = (Conp Γₜ) × (Conp Γₜ)} + {R = λ W → Subp (proj×₁ W) (proj×₂ W)} + {Q = λ W → Subp (Δₚ [ αₜ ]c) (proj×₁ W)} + {eq = ×≡ (≡sym []c-∘) (≡sym []c-∘)} + {f = λ x y → x ∘ₚ (y ∘ₚ αₚ)} + {x = (γₚ [ βₜ ]σₚ) [ αₜ ]σₚ} + {y = βₚ [ αₜ ]σₚ} + ))(substfpoly + {R = λ X → Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) X} + {eq = ≡sym []c-∘} + {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))} + )) + ∘-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 @@ -380,18 +478,11 @@ module FFOLInitial where ,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 - --funlol : {Γₜ Δₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Ξₜ}{σₜ : Subt Γₜ Ξₜ}{δₜ : Subt Δₜ Γₜ}{δₚ : Subp Δₚ (Γₚ [ δₜ ]c)}{A : For Ξₜ}{prf : Pf (con Δₜ (Γₚ [ δₜ ]c)) ((A [ σₜ ∘ₜ δₜ ]f))} → Subp {Δₜ} (Γₚ [ δₜ ]c) ((Ξₚ [ σₜ ∘ₜ δₜ ]c) ▹p⁰ ((A [ σₜ ]f) [ δₜ ]f)) → Subp {Δₜ} (Δₚ) ((Ξₚ [ σₜ ∘ₜ δₜ ]c) ▹p⁰ (A [ σₜ ∘ₜ δₜ ]f)) - --funlol {Γₚ = Γₚ} {Ξₚ = Ξₚ} {σₜ = σₜ} {δₜ = δₜ} {δₚ = δₚ} {prf = prf} (ξ ,ₚ pf) = ((subst (λ X → Subp (Γₚ [ δₜ ]c) ((Ξₚ [ σₜ ∘ₜ δₜ ]c) ▹p⁰ X)) (≡sym []f-∘) ξ) ,ₚ ?) ∘ₚ δₚ - postulate ,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 = Γ} {Δ = Δ} {σ = sub σₜ σₚ} {sub δₜ δₚ} {F = A} {prf} = cong (sub (σₜ ∘ₜ δₜ)) (cong {!funlol!} - (substfpoly - {P = λ X → Subp (Con.p Γ [ δₜ ]c) (X ▹p⁰ ((A [ σₜ ]f) [ δₜ ]f))} - {R = λ X → Subp (Con.p Γ [ δₜ ]c) X} - {eq = ≡sym []c-∘} - {f = λ ξ → ξ ,ₚ (prf [ δₜ ]pₜ)} - {x = σₚ [ δₜ ]σₚ} - )) - -} + ,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}{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})) -- + --_,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 @@ -415,11 +506,11 @@ module FFOLInitial where ; _∘_ = _∘_ ; ∘-ass = ∘-ass ; id = id - ; idl = {!!} - ; idr = {!!} + ; idl = idl + ; idr = idr ; ◇ = ◇ ; ε = sub εₜ εₚ - ; ε-u = {!!} + ; ε-u = cong₂' sub εₜ-u ε-u ; Tm = λ Γ → Tm (Con.t Γ) ; _[_]t = λ t σ → t [ Sub.t σ ]t ; []t-id = []t-id @@ -438,8 +529,8 @@ module FFOLInitial where ; []f-∘ = []f-∘ ; R = r ; R[] = refl - ; _⊢_ = λ Γ A → Pf Γ A - ; _[_]p = λ {Γ}{Δ}{F} pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p + ; _⊢_ = Pf + ; _[_]p = λ pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p ; _▹ₚ_ = _▹p_ ; πₚ¹ = πₚ¹* ; πₚ² = πₚ² diff --git a/PropUtil.agda b/PropUtil.agda index ebada6f..6cae0bf 100644 --- a/PropUtil.agda +++ b/PropUtil.agda @@ -78,10 +78,16 @@ module PropUtil where ≡tran² : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₀ ≡ a₃ ≡tran³ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₀ ≡ a₄ ≡tran⁴ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₀ ≡ a₅ + ≡tran⁵ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ a₆ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₅ ≡ a₆ → a₀ ≡ a₆ + ≡tran⁶ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ a₆ a₇ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₅ ≡ a₆ → a₆ ≡ a₇ → a₀ ≡ a₇ + ≡tran⁷ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₅ ≡ a₆ → a₆ ≡ a₇ → a₇ ≡ a₈ → a₀ ≡ a₈ ≡tran refl refl = refl ≡tran² refl refl refl = refl ≡tran³ refl refl refl refl = refl ≡tran⁴ refl refl refl refl refl = refl + ≡tran⁵ refl refl refl refl refl refl = refl + ≡tran⁶ refl refl refl refl refl refl refl = refl + ≡tran⁷ refl refl refl refl refl refl refl refl = refl cong : {ℓ ℓ' : Level}{A : Set ℓ}{B : Set ℓ'} → (f : A → B) → {a a' : A} → a ≡ a' → f a ≡ f a' cong f refl = refl @@ -93,6 +99,11 @@ module PropUtil where -- We can make a proof-irrelevant substitution substP : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Prop ℓ'){a a' : A} → a ≡ a' → P a → P a' substP P refl h = h + substPP : ∀{ℓ}{A B : Set ℓ}{Q : A → Prop ℓ}{ℓ'}(P : {k : A} → Q k → Prop ℓ'){a a' : A}{x : Q a} + → (eq : a ≡ a') → P x → P (substP Q eq x) + substPP P refl h = h + substP² : ∀{ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Set ℓ'}(P : A → B → Prop ℓ''){a a' : A}{b b' : B} → a ≡ a' → b ≡ b' → P a b → P a' b' + substP² P refl refl p = p postulate coe : ∀{ℓ}{A B : Set ℓ} → A ≡ B → A → B @@ -111,15 +122,25 @@ module PropUtil where coecong f = ≡tran (cong f coerefl) (≡sym coerefl) + coecoe-coe : {ℓ : Level}{A B C : Set ℓ}{eq1 : B ≡ A}{eq2 : C ≡ B}{x : C} → coe eq1 (coe eq2 x) ≡ coe (≡tran eq2 eq1) x + coecoe-coe {eq1 = refl} {refl} = coerefl + coe-f : {ℓ : Level}{A B C D : Set ℓ} → (A → B) → A ≡ C → B ≡ D → C → D coe-f f ac bd x = coe bd (f (coe (≡sym ac) x)) coewtf : {ℓ : Level}{A B C D E F G H : Set ℓ}{ab : A ≡ B}{cd : C ≡ D}{ef : E ≡ F}{gh : G ≡ H}{f : F → B}{g : H → E}{x : G} → {fd : F ≡ D} → f (coe ef (g (coe gh x))) ≡ coe ab ((coe-f f fd (≡sym ab)) (coe cd ((coe-f g (≡sym gh) (≡tran² ef fd (≡sym cd))) x))) coewtf {ab = refl} {refl} {refl} {refl} {f} {g} {fd = refl} = ≡tran (cong f (cong (coe _) (≡sym coeaba))) (≡sym coeaba) + coeshift : {ℓ : Level}{A B : Set ℓ}{x : A} {y : B} {eq : A ≡ B} → coe eq x ≡ y → x ≡ coe (≡sym eq) y + coeshift {eq=refl} refl = ≡sym coeaba + subst : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Set ℓ'){a a' : A} → a ≡ a' → P a → P a' subst P eq p = coe (cong P eq) p - + subst² : ∀{ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Set ℓ'}(P : A → B → Set ℓ''){a a' : A}{b b' : B} → a ≡ a' → b ≡ b' → P a b → P a' b' + subst² P eq eq' p = coe (cong₂ P eq eq') p + subst¹P : ∀{ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Prop ℓ'}(P : A → B → Set ℓ''){a a' : A}{b : B} → a ≡ a' → P a b → P a' b + subst¹P P {b = b} eq p = coe (cong (λ x → P x b) eq) p + --{-# REWRITE transprefl #-} coereflrefl : {ℓ : Level}{A : Set ℓ}{eq eq' : A ≡ A}{a : A} → coe eq (coe eq' a) ≡ a @@ -141,6 +162,26 @@ module PropUtil where {eq : α ≡ β} {f : {ξ : A} → R ξ → P ξ} {x : R α} → coe (cong P eq) (f {α} x) ≡ f (coe (cong R eq) x) substfpoly {eq = refl} {f} = ≡tran coerefl (cong f (≡sym coerefl)) + substppoly : {ℓ ℓ' ℓ'' ℓ''' : Level}{A : Set ℓ}{P : A → Set ℓ'}{R : A → Set ℓ''}{Q : A → Set ℓ'''}{α β : A} + {eq : α ≡ β}{f : {ξ : A} → R ξ → Q ξ → P ξ} {x : R α} {y : Q α} + → coe (cong P eq) (f {α} x y) ≡ f {β} (coe (cong R eq) x) (coe (cong Q eq) y) + substppoly {eq = refl} {f}{x}{y} = ≡tran coerefl (cong₂ f (≡sym coerefl) (≡sym coerefl)) + substfpoly' : {ℓ ℓ' ℓ'' : Level}{A B : Set ℓ}{P R : A → Set ℓ'}{Q : B → Prop ℓ''}{α β : A}{γ δ : B} + {eq : α ≡ β}{eq' : γ ≡ δ} {f : {ξ : A}{ι : B} → R ξ → Q ι → P ξ} {x : R α} {y : Q γ} + → coe (cong P eq) (f {α} {γ} x y) ≡ f {β} {δ} (coe (cong R eq) x) (substP Q eq' y) + substfpoly' {eq = refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl + substfpoly⁴ : {ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{P R : A → Set ℓ'}{Q : A → Prop ℓ''}{α β : A} + {eq : α ≡ β} {f : {ξ : A} → R ξ → Q ξ → P ξ} {x : R α} {y : Q α} + → coe (cong P eq) (f {α} x y) ≡ f {β} (coe (cong R eq) x) (substP Q eq y) + substfpoly⁴ {eq = refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl + substfpoly³ : {ℓ ℓ' ℓ'' ℓ''' : Level}{A B C : Set ℓ}{R : A → Set ℓ'}{Q : B → Prop ℓ''}{P : C → Set ℓ'''}{α β : A}{γ δ : B}{ε φ : C} + {eq : α ≡ β}{eq' : γ ≡ δ}{eq'' : ε ≡ φ} {f : {ξ : A}{ι : B}{τ : C} → R ξ → Q ι → P τ} {x : R α} {y : Q γ} + → coe (cong P eq'') (f {α} {γ} {ε} x y) ≡ f {β} {δ} {φ} (coe (cong R eq) x) (substP Q eq' y) + substfpoly³ {eq = refl} {refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl + substfpoly'' : {ℓ ℓ' ℓ'' : Level}{A C : Set ℓ}{P : A → C → Set ℓ'}{R : A → Set ℓ'}{Q : A → C → Prop ℓ''}{α β : A}{ε φ : C} + {eq : α ≡ β}{eq'' : ε ≡ φ} {f : {ξ : A}{κ : C} → R ξ → Q ξ κ → P ξ κ} {x : R α} {y : Q α ε} + → coe (cong₂ P eq eq'') (f {α} {ε} x y) ≡ f {β} {φ} (coe (cong R eq) x) (substP (λ X → Q X φ) eq (substP (Q α) eq'' y)) + substfpoly'' {eq = refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl substfgpoly : {ℓ ℓ' : Level}{A B : Set ℓ} {P Q : A → Set ℓ'} {R : B → Set ℓ'} {F : B → A} {α β : A} {ε φ : B} {eq₁ : α ≡ β} {eq₂ : F ε ≡ α} {eq₃ : F φ ≡ β} {eq₄ : ε ≡ φ} @@ -150,9 +191,43 @@ module PropUtil where {-# BUILTIN EQUALITY _≡_ #-} + coep² : {ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level} {A : Set ℓ₁} {R : A → Set ℓ₂}{T : A → Set ℓ₃}{S : A → Set ℓ₄}{α β : A} + {p : {ξ : A} → R ξ → T ξ → S ξ}{x : R α}{y : T α}{eq : α ≡ β} + → subst S (≡sym eq) (p {β} (subst R eq x) (subst T eq y)) ≡ p {α} x y + coep² {S = S}{p = p}{x}{y}{refl} = ≡tran (substrefl {P = S} {e = refl}) (cong₂ p (substrefl {a = x} {e = refl}) (substrefl {a = y} {e = refl})) + coep²'' : {ℓ ℓ' : Level} {A : Set ℓ} {R S : A → Set ℓ'}{T : A → Prop ℓ'}{α β : A} + {p : {ξ : A} → R ξ → T ξ → S ξ}{x : R α}{y : T α}{eq : α ≡ β} + → subst S (≡sym eq) (p {β} (subst R eq x) (substP T eq y)) ≡ p {α} x y + coep²'' {S = S}{p = p}{x}{y}{refl} = ≡tran (substrefl {P = S} {e = refl}) (cong (λ X → p X y) (substrefl {a = x} {e = refl})) + coep²' : {ℓ ℓ' : Level} {A : Set ℓ} {R T S : A → Set ℓ'}{α β : A} + {p : {ξ : A} → R ξ → T ξ → S ξ}{x : R β}{y : T α}{eq : α ≡ β} + → subst S (≡sym eq) (p {β} x (subst T eq y)) ≡ p {α} (subst R (≡sym eq) x) y + coep²' {S = S}{p = p}{x}{y}{refl} = ≡tran (substrefl {P = S} {e = refl}) (cong₂ p (≡sym (substrefl {a = x} {e = refl})) (substrefl {a = y} {e = refl})) - - + coep∘ : {ℓ ℓ' : Level}{A : Set ℓ} {R : A → A → Set ℓ'} {α β γ δ ε φ : A} + {p : {x y z : A} → R x y → R z x → R z y}{x : R β γ}{y : R α β} + {eq1 : α ≡ δ} {eq2 : β ≡ ε} {eq3 : γ ≡ φ} → + coe (cong₂ R (≡sym eq1) (≡sym eq3)) (p (coe (cong₂ R eq2 eq3) x) (coe (cong₂ R eq1 eq2) y)) ≡ p x y + coep∘ {p = p}{eq1 = refl}{refl}{refl} = ≡tran coerefl (cong₂ p coerefl coerefl) + coep∘-helper = λ {ℓ ℓ' ℓ'' : Level}{B : Set ℓ}{A : B → Set ℓ''} {R : (b : B) → A b → A b → Set ℓ'} + {b₁ b₂ : B} {α γ : A b₁} {δ φ : A b₂} + {eq0 : b₁ ≡ b₂}{eqa : subst A eq0 α ≡ δ}{eqb : subst A eq0 γ ≡ φ} + → (≡tran² (cong (R b₂ δ) (≡sym eqb)) (cong (λ X → R b₂ X (subst A eq0 γ)) (≡sym eqa)) (≡tran (≡sym (substrefl {P = λ X → Set ℓ'}{a = R b₂ (subst A eq0 α) (subst A eq0 γ)}{e = refl})) (coep² {p = λ {t} x y → R t x y}{eq = eq0}))) + coep∘-helper-coe : {ℓ ℓ' ℓ'' : Level}{B : Set ℓ}{A : B → Set ℓ''} {R : (b : B) → A b → A b → Set ℓ'} + {b₁ b₂ : B} {α γ : A b₁} {δ φ : A b₂} + {eq0 : b₁ ≡ b₂}{eqa : subst A eq0 α ≡ δ}{eqb : subst A eq0 γ ≡ φ} → {a : R b₂ δ φ}{a' : R b₁ α γ} → coe (coep∘-helper {eq0 = eq0} {eqa = eqa} {eqb = eqb}) a ≡ a + coep∘-helper-coe {eq0 = refl}{refl}{refl} = coerefl + {-coep∘' : {ℓ ℓ' ℓ'' : Level}{B : Set ℓ}{A : B → Set ℓ''} {R : (b : B) → A b → A b → Set ℓ'} + {b₁ b₂ : B} {α β γ : A b₁} {δ ε φ : A b₂} + {p : {b : B}{x y z : A b} → R b x y → R b z x → R b z y}{x : R b₁ β γ}{y : R b₁ α β} + {eq0 : b₁ ≡ b₂}{eq1 : subst A eq0 α ≡ δ} {eq2 : subst A eq0 β ≡ ε} {eq3 : subst A eq0 γ ≡ φ} + {eq4 : R b₂ δ φ ≡ R b₁ α γ}{eq5 : R b₂ ε φ ≡ R b₁ β γ}{eq6 : R b₂ δ ε ≡ R b₁ α β} + → coe eq4 + (p {b₂} {ε} {φ} {δ} (coe (≡sym (eq5)) x) (coe (≡sym ( + eq6 + )) y)) ≡ p {b₁} {β} {γ} {α} x y + --coep∘' {p = p} {x} {y} {eq0 = refl} {refl} {refl} {refl} {eq4} = {!!} + -} @@ -184,6 +259,13 @@ module PropUtil where a : A b : B a + record _×ᵈ_ (A : Set ℓ) (B : A → Set ℓ') : Set (ℓ ⊔ ℓ') where + constructor _,×ᵈ_ + field + a : A + b : B a + + proj×₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : Set ℓ'} → (A × B) → A proj×₁ p = _×_.a p proj×₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : Set ℓ'} → (A × B) → B @@ -199,4 +281,14 @@ module PropUtil where proj×''₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Prop ℓ'} → (p : A ×'' B) → B (proj×''₁ p) proj×''₂ p = _×''_.b p + proj×ᵈ₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Set ℓ'} → (A ×ᵈ B) → A + proj×ᵈ₁ p = _×ᵈ_.a p + proj×ᵈ₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Set ℓ'} → (p : A ×ᵈ B) → (B (proj×ᵈ₁ p)) + proj×ᵈ₂ p = _×ᵈ_.b p + + ×≡ : {A : Set ℓ}{B : Set ℓ'}{a a' : A}{b b' : B} → a ≡ a' → b ≡ b' → a ,× b ≡ a' ,× b' + ×≡ refl refl = refl + + ×ᵈ≡ : {A : Set ℓ}{B : A → Set ℓ'}{a a' : A}{b : B a}{b' : B a'} → (eq : a ≡ a') → subst B eq b ≡ b' → a ,×ᵈ b ≡ a' ,×ᵈ b' + ×ᵈ≡ {B = B} {a = a}{b = b} refl refl = cong₂' _,×ᵈ_ refl refl