Tried to add a completeness proof for zol
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@ -244,6 +244,12 @@ module PropUtil where
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a : A
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b : B
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record _×p_ (A : Prop ℓ) (B : A → Prop ℓ') : Prop (ℓ ⊔ ℓ') where
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constructor _,×p_
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field
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a : A
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b : B a
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record _×''_ (A : Set ℓ) (B : A → Prop ℓ') : Set (ℓ ⊔ ℓ') where
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constructor _,×''_
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field
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@ -267,6 +273,11 @@ module PropUtil where
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proj×'₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : Prop ℓ'} → (A ×' B) → B
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proj×'₂ p = _×'_.b p
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proj×p₁ : {ℓ ℓ' : Level}{A : Prop ℓ}{B : A → Prop ℓ'} → (A ×p B) → A
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proj×p₁ p = _×p_.a p
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proj×p₂ : {ℓ ℓ' : Level}{A : Prop ℓ}{B : A → Prop ℓ'} → (p : A ×p B) → B (proj×p₁ p)
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proj×p₂ p = _×p_.b p
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proj×''₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Prop ℓ'} → (A ×'' B) → A
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proj×''₁ p = _×''_.a p
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proj×''₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Prop ℓ'} → (p : A ×'' B) → B (proj×''₁ p)
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19
ZOL2.lagda
19
ZOL2.lagda
@ -9,7 +9,8 @@ module ZOL2 where
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open import ListUtil
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variable
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ℓ¹ ℓ² ℓ³ ℓ⁴ ℓ⁵ : Level
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ℓ¹ ℓ² ℓ³ ℓ⁴ : Level
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ℓ¹' ℓ²' ℓ³' ℓ⁴' : Level
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record ZOL : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴)) where
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infixr 10 _∘_
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@ -77,7 +78,7 @@ module ZOL2 where
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lam : {Γ : Con}{F G : For Γ} → Pf (Γ ▹ₚ F) (G [ πₚ¹ id ]f) → Pf Γ (F ⇒ G)
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app : {Γ : Con}{F G : For Γ} → Pf Γ (F ⇒ G) → Pf Γ F → Pf Γ G
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record Mapping (S : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}) (D : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴)) where
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record Mapping (S : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}) (D : ZOL {ℓ¹'} {ℓ²'} {ℓ³'} {ℓ⁴'}) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ¹' ⊔ ℓ²' ⊔ ℓ³' ⊔ ℓ⁴')) where
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field
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-- We first make the base category with its final object
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@ -87,7 +88,7 @@ module ZOL2 where
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mPf : {Γ : (ZOL.Con S)} {A : ZOL.For S Γ} → ZOL.Pf S Γ A → ZOL.Pf D (mCon Γ) (mFor A)
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record Morphism (S : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}) (D : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴)) where
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record Morphism (S : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}) (D : ZOL {ℓ¹'} {ℓ²'} {ℓ³'} {ℓ⁴'}) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ¹' ⊔ ℓ²' ⊔ ℓ³' ⊔ ℓ⁴')) where
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field m : Mapping S D
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mCon = Mapping.mCon m
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mSub = Mapping.mSub m
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@ -101,4 +102,16 @@ module ZOL2 where
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e⇒ : {Γ : ZOL.Con S}{A B : ZOL.For S Γ} → mFor (ZOL._⇒_ S A B) ≡ ZOL._⇒_ D (mFor A) (mFor B)
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-- No equation needed for lam, app, ∀i, ∀e as their output are in prop
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record TrNat {S : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}} {D : ZOL {ℓ¹'} {ℓ²'} {ℓ³'} {ℓ⁴'}} (a : Mapping S D) (b : Mapping S D) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ¹' ⊔ ℓ²' ⊔ ℓ³' ⊔ ℓ⁴')) where
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field
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f : (Γ : ZOL.Con S) → ZOL.Sub D (Mapping.mCon a Γ) (Mapping.mCon b Γ)
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-- Unneeded because Sub are in prop
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--eq : (Γ Δ : ZOL.Con S)(σ : ZOL.Sub S Γ Δ) → (ZOL._∘_ D (f Δ) (Mapping.mSub a σ)) ≡ (ZOL._∘_ D (Mapping.mSub b σ) (f Γ))
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_∘TrNat_ : {S : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}}{D : ZOL {ℓ¹'} {ℓ²'} {ℓ³'} {ℓ⁴'}}{a b c : Mapping S D} → TrNat a b → TrNat b c → TrNat a c
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_∘TrNat_ {D = D} α β = record { f = λ Γ → ZOL._∘_ D (TrNat.f β Γ) (TrNat.f α Γ) }
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idTrNat : {S : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}}{D : ZOL {ℓ¹'} {ℓ²'} {ℓ³'} {ℓ⁴'}}{a : Mapping S D} → TrNat a a
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idTrNat {D = D} = record { f = λ Γ → ZOL.id D }
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\end{code}
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198
ZOLCompleteness.lagda
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198
ZOLCompleteness.lagda
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@ -0,0 +1,198 @@
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\begin{code}
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{-# OPTIONS --prop --rewriting #-}
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open import PropUtil
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module ZOLCompleteness where
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open import Agda.Primitive
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open import ZOL2
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open import ListUtil
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record Kripke : Set (lsuc (ℓ¹)) where
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field
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World : Set ℓ¹
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_-w->_ : World → World → Prop ℓ¹ -- arrows
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-w->id : {w : World} → w -w-> w -- id arrow
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_∘-w->_ : {w w' w'' : World} → w -w-> w' → w' -w-> w'' → w -w-> w'' -- arrow composition
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Ι : World → Prop ℓ¹
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Ι≤ : {w w' : World} → w -w-> w' → Ι w' → Ι w
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infixr 10 _∘_
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Con : Set (lsuc ℓ¹)
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Con = World → Prop ℓ¹
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Sub : Con → Con → Prop ℓ¹
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Sub Δ Γ = (w : World) → Δ w → Γ w
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_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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α ∘ β = λ w γ → α w (β w γ)
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id : {Γ : Con} → Sub Γ Γ
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id = λ w γ → γ
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◇ : Con -- The initial object of the category
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◇ = λ w → ⊤
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε w Γ = tt
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-- Functor Con → Set called For
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For : Con → Set (lsuc ℓ¹)
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For Γ = (w : World) → (Γ w) → Prop ℓ¹
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_[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms
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F [ σ ]f = λ w → λ x → F w (σ w x)
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-- Proofs
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Pf : (Γ : Con) → For Γ → Prop ℓ¹
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Pf Γ F = ∀ w (γ : Γ w) → F w γ
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_[_]p : {Γ Δ : Con} → {F : For Γ} → Pf Γ F → (σ : Sub Δ Γ) → Pf Δ (F [ σ ]f) -- The functor's action on morphisms
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prf [ σ ]p = λ w → λ γ → prf w (σ w γ)
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-- Equalities below are useless because Γ ⊢ F is in prop
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-- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf
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-- []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
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-- → Prop⁺
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_▹ₚ_ : (Γ : Con) → For Γ → Con
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Γ ▹ₚ F = λ w → Γ w ×p F w
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πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
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πₚ¹ σ w δ = proj×p₁ (σ w δ)
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πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Pf Δ (F [ πₚ¹ σ ]f)
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πₚ² σ w δ = proj×p₂ (σ w δ)
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_,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Pf Δ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F)
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(σ ,ₚ pf) w δ = (σ w δ) ,×p pf w δ
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-- Base formula
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ι : {Γ : Con} → For Γ
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ι = λ w → λ γ → Ι w
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-- Implication
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_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
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(F ⇒ G) w γ = {w' : World} → (s : w -w-> w') → (F w' {!s γ!}) → (G w' {!!})
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-- Lam & App
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lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → Pf (Γ ▹ₚ F) (G [ πₚ¹ id ]f) → Pf Γ (F ⇒ G)
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--lam prf = λ w γ w' s h → prf w (γ ,×p h)
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app : {Γ : Con} → {F G : For Γ} → Pf Γ (F ⇒ G) → Pf Γ F → Pf Γ G
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--app prf prf' = λ w γ → prf w γ w -w->id (prf' w γ)
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-- Again, we don't write the _[_]p equalities as everything is in Prop
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zol : ZOL
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zol = record
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{ Con = Con
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; Sub = Sub
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; _∘_ = _∘_
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; id = id
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; ◇ = ◇
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; ε = ε
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; For = For
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; _[_]f = _[_]f
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; []f-id = refl
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; []f-∘ = refl
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; Pf = Pf
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; _[_]p = _[_]p
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; _▹ₚ_ = _▹ₚ_
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; πₚ¹ = πₚ¹
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; πₚ² = πₚ²
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; _,ₚ_ = _,ₚ_
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; ι = ι
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; []f-ι = refl
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; _⇒_ = _⇒_
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; []f-⇒ = refl
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; lam = lam
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; app = app
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}
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\end{code}
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module U where
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import ZOLInitial as I
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U : Kripke
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U = record
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{ World = I.Con
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; _-w->_ = I.Sub
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; -w->id = I.id
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; _∘-w->_ = λ σ σ' → σ' I.∘ σ
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; Ι = λ Γ → I.Pf Γ I.ι
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; Ι≤ = λ s pf → pf I.[ s ]p
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}
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open Kripke U
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y : Mapping I.zol zol
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y = record
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{ mCon = λ Γ Δ → I.Sub Δ Γ
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; mSub = λ σ Ξ δ → σ I.∘ δ
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; mFor = λ A Ξ σ → I.Pf Ξ A
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; mPf = λ pf Ξ σ → pf I.[ σ ]p
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}
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m : Morphism I.zol zol
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m = I.InitialMorphism.mor zol
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u : (Γ : I.Con) → Sub (Morphism.mCon m Γ) (Mapping.mCon y Γ)
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q : (Γ : I.Con) → Sub (Mapping.mCon y Γ) (Morphism.mCon m Γ)
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⟦_⟧c = Morphism.mCon m
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⟦_,_⟧f = λ A Γ → Morphism.mFor m {Γ} A
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⟦⟧f-mon : {Γ : I.Con}{A : I.For}{w w' : World}{γ : ⟦ Γ ⟧c w}{γ' : ⟦ Γ ⟧c w'} → w -w-> w' → ⟦ A , Γ ⟧f w' γ' → ⟦ A , Γ ⟧f w γ
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⟦⟧f-mon {A = I.ι} s h = Ι≤ s h
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⟦⟧f-mon {A = A I.⇒ B} s h w'' s' h' = {!h!}
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⟦⟧c-mon : {Γ : I.Con}{w w' : I.Con} → w -w-> w' → ⟦ Γ ⟧c w → ⟦ Γ ⟧c w'
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⟦⟧c-mon s h = {!!}
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q⁰ : {F : I.For} → {Γ : I.Con} → Pf ⟦ Γ ⟧c ⟦ F , Γ ⟧f → I.Pf Γ F
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u⁰ : {F : I.For} → {Γ : I.Con} → I.Pf Γ F → Pf ⟦ Γ ⟧c ⟦ F , Γ ⟧f
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u⁰ {I.ι} h w γ = {!!}
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u⁰ {A I.⇒ B} h Γ' γ Γ'' iq hF = u⁰ {B} (I.app {A = A} h (q⁰ (λ Ξ γ' → {!hF!}))) {!!} γ --{Γ'} iq hF = u {F₁} (app {Γ'} {F} {F₁} (⊢tran iq h) (q hF))
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q⁰ {I.ι} {Γ} h = h Γ {!!}
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q⁰ {A I.⇒ B} {Γ} h = I.lam (q⁰ (λ w γ → {!h ? ? ? (I.πₚ¹ I.id)!})) --lam (q (h (retro (Preorder.refl≤ o)) (u {F} {F ∷ Γ} zero)))
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u I.◇ Δ x = I.ε
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u (Γ I.▹ₚ I.ι) Δ (Γu ,×p Au) = u Γ Δ Γu I.,ₚ Au
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--u (Γ I.▹ₚ (A I.⇒ B)) Δ (Γu ,×p ABu) = (u Γ Δ Γu) I.,ₚ I.lam (I.πₚ² (u (Γ I.▹ₚ B) (Δ I.▹ₚ A) ({!!} ,×p {!!})))
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u (Γ I.▹ₚ (A I.⇒ B)) Δ (Γu ,×p ABu) = (u Γ Δ Γu) I.,ₚ {!!}
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q .I.◇ Δ I.ε = tt
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q (Γ I.▹ₚ I.ι) Δ σ = (q Γ Δ (I.πₚ¹ σ)) ,×p I.πₚ² σ
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q (Γ I.▹ₚ (A I.⇒ B)) Δ σ = (q Γ Δ (I.πₚ¹ σ)) ,×p {!!}
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{-
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u {Var x} h = h
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u {F ⇒ F₁} h {Γ'} iq hF = u {F₁} (app {Γ'} {F} {F₁} (⊢tran iq h) (q hF))
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u {F ∧∧ G} h = ⟨ (u {F} (ande₁ h)) , (u {G} (ande₂ h)) ⟩
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u {⊤⊤} h = tt
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q {Var x} h = neu⁰ h
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q {F ⇒ F₁} {Γ} h = lam (q (h (retro (Preorder.refl≤ o)) (u {F} {F ∷ Γ} zero)))
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q {F ∧∧ G} ⟨ hF , hG ⟩ = andi (q {F} hF) (q {G} hG)
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q {⊤⊤} h = true
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-}
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ηu : TrNat (Morphism.m m) y
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ηu = record { f = u }
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ηq : TrNat y (Morphism.m m)
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ηq = record { f = q }
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eq : ηu ∘TrNat ηq ≡ idTrNat
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eq = {!!}
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-- Transformation naturelle
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\end{code}
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-- Completeness proof
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-- We first build our universal Kripke model
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module ComplenessProof where
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-- We have a model, we construct the Universal Presheaf model of this model
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import ZOLInitial as I
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\end{code}
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@ -109,7 +109,7 @@ module ZOLInitial where
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; app = app
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}
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module InitialMorphism (M : ZOL {lzero} {lzero} {lzero} {lzero}) where
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module InitialMorphism (M : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}) where
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mCon : Con → (ZOL.Con M)
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mFor : {Γ : Con} → For → (ZOL.For M (mCon Γ))
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@ -156,7 +156,7 @@ module ZOLInitial where
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; e⇒ = refl
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}
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module InitialMorphismUniqueness {M : ZOL {lzero} {lzero} {lzero} {lzero}} {m : Morphism zol M} where
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module InitialMorphismUniqueness {M : ZOL {ℓ¹} {ℓ²} {ℓ³} {ℓ⁴}} {m : Morphism zol M} where
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open InitialMorphism M
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mCon≡ : {Γ : Con} → mCon Γ ≡ (Morphism.mCon m Γ)
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@ -61,7 +61,8 @@ module ZOLNormalization (PV : Set) where
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u : {F : Form} → {Γ : Con} → Γ ⊢⁰ F → Γ ⊩ᶠ F
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u {Var x} h = h
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u {F ⇒ F₁} h {Γ'} iq hF = u {F₁} (app {Γ'} {F} {F₁} (⊢tran iq h) (q hF))
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u {F ⇒ F₁} h {Γ'} iq hF = {!!}
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--u {F ⇒ F₁} h {Γ'} iq hF = u {F₁} (app {Γ'} {F} {F₁} (⊢tran iq h) (q hF))
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u {F ∧∧ G} h = ⟨ (u {F} (ande₁ h)) , (u {G} (ande₂ h)) ⟩
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u {⊤⊤} h = tt
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