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Finished adding conjunction.

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Mysaa 2023-06-01 12:26:15 +02:00
parent e1f1ea32f3
commit 01a029bc39
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3 changed files with 34 additions and 10 deletions
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14
PropositionalKripke.agda
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@ -64,9 +64,9 @@ module PropositionalKripke (PV : Set) where
⟦ zero (next∈ h) ⟧ wΓ = ⟦ zero h ⟧ (proj₂ wΓ)
⟦ lam p ⟧ = λ wΓ w≤ w'A → ⟦ p ⟧ ⟨ w'A , mon⊩ᶜ w≤ wΓ ⟩
⟦ app p p₁ ⟧ wΓ = ⟦ p ⟧ wΓ refl≤ (⟦ p₁ ⟧ wΓ)
⟦ andi p₁ p₂ ⟧ = {!!}
⟦ ande₁ p ⟧ = {!!}
⟦ ande₂ p ⟧ = {!!}
⟦ andi p₁ p₂ ⟧ wΓ = ⟨ (⟦ p₁ ⟧ wΓ) , (⟦ p₂ ⟧ wΓ) ⟩
⟦ ande₁ p ⟧ wΓ = proj₁ $ ⟦ p ⟧ wΓ
⟦ ande₂ p ⟧ wΓ = proj₂ $ ⟦ p ⟧ wΓ
@ -91,10 +91,10 @@ module PropositionalKripke (PV : Set) where
⊢→⊩ᶠ : {F : Form} → {Γ : Con} → Γ ⊢ F → Γ ⊩ᶠ F
⊢→⊩ᶠ {Var x} h = h
⊢→⊩ᶠ {F ⇒ F₁} h {Γ'} iq hF = ⊢→⊩ᶠ {F₁} (app {Γ'} {F} {F₁} (lam (app (halftran⊢⁺ (addhyp⊢⁺ (right∈* refl∈*) iq) h) (zero zero∈))) (⊩ᶠ→⊢ hF))
⊢→⊩ᶠ {F ∧∧ G} h = {!!}
⊢→⊩ᶠ {F ∧∧ G} h = ⟨ (⊢→⊩ᶠ {F} (ande₁ h)) , (⊢→⊩ᶠ {G} (ande₂ h)) ⟩
⊩ᶠ→⊢ {Var x} h = h
⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁺ (right∈* refl∈*) refl⊢⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} (zero zero∈))))
⊩ᶠ→⊢ {F ∧∧ G} ⟨ hF , hG ⟩ = {!!}
⊩ᶠ→⊢ {F ∧∧ G} ⟨ hF , hG ⟩ = andi (⊩ᶠ→⊢ {F} hF) (⊩ᶠ→⊢ {G} hG)
-- Finally, we can deduce completeness of the Kripke model
completeness : {F : Form} → [] ⊫ F → [] ⊢ F
@ -124,10 +124,10 @@ module PropositionalKripke (PV : Set) where
⊢→⊩ᶠ {Var x} h = h
⊢→⊩ᶠ {F ⇒ F₁} h {Γ'} iq hF = ⊢→⊩ᶠ {F₁} (app {Γ'} {F} {F₁} (halftran⊢⁰⁺⁰ iq h) (⊩ᶠ→⊢ hF))
⊢→⊩ᶠ {F ∧∧ G} h = ?
⊢→⊩ᶠ {F ∧∧ G} h = ⟨ (⊢→⊩ᶠ {F} (ande₁ h)) , (⊢→⊩ᶠ {G} (ande₂ h)) ⟩
⊩ᶠ→⊢ {Var x} h = neu⁰ h
⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁰⁺ (right∈* refl∈*) refl⊢⁰⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} (zero zero∈))))
⊩ᶠ→⊢ {F ∧∧ G} h = {!!}
⊩ᶠ→⊢ {F ∧∧ G} ⟨ hF , hG ⟩ = andi (⊩ᶠ→⊢ {F} hF) (⊩ᶠ→⊢ {G} hG)
module OtherProofs where

18
PropositionalKripkeGeneral.agda
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@ -4,7 +4,7 @@ module PropositionalKripkeGeneral (PV : Set) where
open import ListUtil
open import PropUtil
open import PropositionalLogic PV using (Form; Var; _⇒_; Con)
open import PropositionalLogic PV using (Form; Var; _⇒_; _∧∧_; Con)
open import PropositionalKripke PV using (Kripke)
@ -24,8 +24,11 @@ module PropositionalKripkeGeneral (PV : Set) where
_⊢*_ : Con → Form → Prop
zero : {Γ : Con} → {F : Form} → (F ∷ Γ) ⊢⁰ F
app : {Γ : Con} → {F G : Form} → Γ ⊢⁰ (F ⇒ G) → Γ ⊢* F → Γ ⊢⁰ G
ande₁ : {Γ : Con} → {F G : Form} → Γ ⊢⁰ (F ∧∧ G) → Γ ⊢⁰ F
ande₂ : {Γ : Con} → {F G : Form} → Γ ⊢⁰ (F ∧∧ G) → Γ ⊢⁰ G
neu⁰ : {Γ : Con} → {x : PV} → Γ ⊢⁰ Var x → Γ ⊢* Var x
lam : {Γ : Con} → {F G : Form} → (F ∷ Γ) ⊢* G → Γ ⊢* (F ⇒ G)
andi : {Γ : Con} → {F G : Form} → Γ ⊢* F → Γ ⊢* G → Γ ⊢* (F ∧∧ G)
record NormalizationFrame : Set₁ where
field
@ -58,8 +61,11 @@ module PropositionalKripkeGeneral (PV : Set) where
u {Var x} h = h
u {F ⇒ F₁} h {Γ'} iq hF = u {F₁} (app {Γ'} {F} {F₁} (⊢tran iq h) (q hF))
u {F ∧∧ G} h = ⟨ (u {F} (ande₁ h)) , (u {G} (ande₂ h)) ⟩
q {Var x} h = neu⁰ h
q {F ⇒ F₁} {Γ} h = lam (q (h (retro (Preorder.refl≤ o)) (u {F} {F ∷ Γ} zero)))
q {F ∧∧ G} ⟨ hF , hG ⟩ = andi (q {F} hF) (q {G} hG)
@ -77,8 +83,11 @@ module PropositionalKripkeGeneral (PV : Set) where
_⊢*_ = _⊢*_ ;
zero = zero zero∈ ;
app = app ;
ande₁ = ande₁;
ande₂ = ande₂ ;
neu⁰ = neu⁰ ;
lam = lam
lam = lam;
andi = andi
}
BiggestNN : NormalAndNeutral
@ -88,8 +97,11 @@ module PropositionalKripkeGeneral (PV : Set) where
_⊢*_ = _⊢_ ;
zero = zero zero∈ ;
app = app ;
ande₁ = ande₁ ;
ande₂ = ande₂ ;
neu⁰ = λ x → x ;
lam = lam
lam = lam ;
andi = andi
}
PO⊢⁺ = [ order {Con} _⊢⁺_ refl⊢⁺ tran⊢⁺ ]ᵒᵖ

12
PropositionalLogic.agda
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@ -109,9 +109,12 @@ module PropositionalLogic (PV : Set) where
data _⊢⁰_ : Con → Form → Prop where
zero : A ∈ Γ → Γ ⊢⁰ A
app : Γ ⊢⁰ (A ⇒ B) → Γ ⊢* A → Γ ⊢⁰ B
ande₁ : Γ ⊢⁰ A ∧∧ B → Γ ⊢⁰ A
ande₂ : Γ ⊢⁰ A ∧∧ B → Γ ⊢⁰ B
data _⊢*_ : Con → Form → Prop where
neu⁰ : Γ ⊢⁰ Var x → Γ ⊢* Var x
lam : (A ∷ Γ) ⊢* B → Γ ⊢* (A ⇒ B)
andi : Γ ⊢* A → Γ ⊢* B → Γ ⊢* (A ∧∧ B)
infix 5 _⊢⁰_
infix 5 _⊢*_
@ -120,16 +123,22 @@ module PropositionalLogic (PV : Set) where
⊢*→⊢ : Γ ⊢* F → Γ ⊢ F
⊢⁰→⊢ (zero h) = zero h
⊢⁰→⊢ (app h x) = app (⊢⁰→⊢ h) (⊢*→⊢ x)
⊢⁰→⊢ (ande₁ h) = ande₁ (⊢⁰→⊢ h)
⊢⁰→⊢ (ande₂ h) = ande₂ (⊢⁰→⊢ h)
⊢*→⊢ (neu⁰ x) = ⊢⁰→⊢ x
⊢*→⊢ (lam h) = lam (⊢*→⊢ h)
⊢*→⊢ (andi h₁ h₂) = andi (⊢*→⊢ h₁) (⊢*→⊢ h₂)
-- We can add hypotheses to a proof
addhyp⊢⁰ : Γ ∈* Γ' → Γ ⊢⁰ A → Γ' ⊢⁰ A
addhyp⊢* : Γ ∈* Γ' → Γ ⊢* A → Γ' ⊢* A
addhyp⊢⁰ s (zero x) = zero (mon∈∈* x s)
addhyp⊢⁰ s (app h h₁) = app (addhyp⊢⁰ s h) (addhyp⊢* s h₁)
addhyp⊢⁰ s (ande₁ h) = ande₁ (addhyp⊢⁰ s h)
addhyp⊢⁰ s (ande₂ h) = ande₂ (addhyp⊢⁰ s h)
addhyp⊢* s (neu⁰ x) = neu⁰ (addhyp⊢⁰ s x)
addhyp⊢* s (lam h) = lam (addhyp⊢* (both∈* s) h)
addhyp⊢* s (andi h₁ h₂) = andi (addhyp⊢* s h₁) (addhyp⊢* s h₂)
-- Extension of ⊢⁰ to contexts
-- i.e. there is a neutral proof for any element
@ -163,9 +172,12 @@ module PropositionalLogic (PV : Set) where
halftran⊢⁰⁺⁰ : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁰⁺ Γ' → Γ' ⊢⁰ F → Γ ⊢⁰ F
halftran⊢⁰⁺* h⁺ (neu⁰ x) = neu⁰ (halftran⊢⁰⁺⁰ h⁺ x)
halftran⊢⁰⁺* h⁺ (lam h) = lam (halftran⊢⁰⁺* ⟨ zero zero∈ , addhyp⊢⁰⁺ (right∈* refl∈*) h⁺ ⟩ h)
halftran⊢⁰⁺* h⁺ (andi h₁ h₂) = andi (halftran⊢⁰⁺* h⁺ h₁) (halftran⊢⁰⁺* h⁺ h₂)
halftran⊢⁰⁺⁰ {Γ' = x ∷ Γ'} h⁺ (zero zero∈) = proj₁ h⁺
halftran⊢⁰⁺⁰ {Γ' = x ∷ Γ'} h⁺ (zero (next∈ h)) = halftran⊢⁰⁺⁰ (proj₂ h⁺) (zero h)
halftran⊢⁰⁺⁰ h⁺ (app h h') = app (halftran⊢⁰⁺⁰ h⁺ h) (halftran⊢⁰⁺* h⁺ h')
halftran⊢⁰⁺⁰ h⁺ (ande₁ h) = ande₁ (halftran⊢⁰⁺⁰ h⁺ h)
halftran⊢⁰⁺⁰ h⁺ (ande₂ h) = ande₂ (halftran⊢⁰⁺⁰ h⁺ h)
-- The relation is transitive
tran⊢⁰⁺ : {Γ Γ' Γ'' : Con} → Γ ⊢⁰⁺ Γ' → Γ' ⊢⁰⁺ Γ'' → Γ ⊢⁰⁺ Γ''