Added normalization proof in a separate file
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Normalization.agda
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135
Normalization.agda
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@ -0,0 +1,135 @@
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{-# OPTIONS --prop #-}
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module Normalization (PV : Set) where
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open import PropUtil
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open import PropositionalLogic PV
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open import PropositionalKripke PV
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private
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variable
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A : Form
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B : Form
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F : Form
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G : Form
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Γ : Con
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Γ' : Con
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x : PV
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-- ⊢⁰ are neutral forms
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-- ⊢* are normal forms
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mutual
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data _⊢⁰_ : Con → Form → Prop where
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zero : (A ∷ Γ) ⊢⁰ A
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next : Γ ⊢⁰ A → (B ∷ Γ) ⊢⁰ A
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app : Γ ⊢⁰ (A ⇒ B) → Γ ⊢* A → Γ ⊢⁰ B
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data _⊢*_ : Con → Form → Prop where
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neu⁰ : Γ ⊢⁰ Var x → Γ ⊢* Var x
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lam : (A ∷ Γ) ⊢* B → Γ ⊢* (A ⇒ B)
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⊢⁰→⊢ : Γ ⊢⁰ F → Γ ⊢ F
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⊢*→⊢ : Γ ⊢* F → Γ ⊢ F
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⊢⁰→⊢ zero = zero
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⊢⁰→⊢ (next h) = next (⊢⁰→⊢ h)
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⊢⁰→⊢ (app h x) = app (⊢⁰→⊢ h) (⊢*→⊢ x)
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⊢*→⊢ (neu⁰ x) = ⊢⁰→⊢ x
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⊢*→⊢ (lam h) = lam (⊢*→⊢ h)
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private
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data _⊆_ : Con → Con → Prop where
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zero⊆ : Γ ⊆ Γ
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next⊆ : Γ ⊆ Γ' → Γ ⊆ (A ∷ Γ')
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retro⊆ : {Γ Γ' : Con} → {A : Form} → (A ∷ Γ) ⊆ Γ' → Γ ⊆ Γ'
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retro⊆ {Γ' = []} () -- Impossible to have «A∷Γ ⊆ []»
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retro⊆ {Γ' = x ∷ Γ'} zero⊆ = next⊆ zero⊆
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retro⊆ {Γ' = x ∷ Γ'} (next⊆ h) = next⊆ (retro⊆ h)
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-- Extension of ⊢⁰ to contexts
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_⊢⁺⁰_ : Con → Con → Prop
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Γ ⊢⁺⁰ [] = ⊤
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Γ ⊢⁺⁰ (F ∷ Γ') = (Γ ⊢⁰ F) ∧ (Γ ⊢⁺⁰ Γ')
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infix 5 _⊢⁺⁰_
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-- This relation is reflexive
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private -- Lemma showing that the relation respects ⊆
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mon⊆≤⁰ : Γ' ⊆ Γ → Γ ⊢⁺⁰ Γ'
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mon⊆≤⁰ {[]} sub = tt
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mon⊆≤⁰ {x ∷ Γ} zero⊆ = ⟨ zero , mon⊆≤⁰ (next⊆ zero⊆) ⟩
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mon⊆≤⁰ {x ∷ Γ} (next⊆ sub) = ⟨ (next (proj₁ (mon⊆≤⁰ sub)) ) , mon⊆≤⁰ (next⊆ (retro⊆ sub)) ⟩
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refl⊢⁺⁰ : Γ ⊢⁺⁰ Γ
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refl⊢⁺⁰ {[]} = tt
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refl⊢⁺⁰ {x ∷ Γ} = ⟨ zero , mon⊆≤⁰ (next⊆ zero⊆) ⟩
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-- A useful lemma, that we can add hypotheses
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addhyp⊢⁺⁰ : Γ ⊢⁺⁰ Γ' → (A ∷ Γ) ⊢⁺⁰ Γ'
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addhyp⊢⁺⁰ {Γ' = []} h = tt
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addhyp⊢⁺⁰ {Γ' = A ∷ Γ'} h = ⟨ next (proj₁ h) , addhyp⊢⁺⁰ (proj₂ h) ⟩
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{- We use a slightly different Universal Kripke Model -}
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module UniversalKripke⁰ where
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Worlds = Con
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_≤_ : Con → Con → Prop
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Γ ≤ Η = Η ⊢⁺⁰ Γ
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_⊩_ : Con → PV → Prop
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Γ ⊩ x = Γ ⊢⁰ Var x
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refl≤ = refl⊢⁺⁰
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-- Proving transitivity
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halftran≤* : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁺⁰ Γ' → Γ' ⊢* F → Γ ⊢* F
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halftran≤⁰ : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁺⁰ Γ' → Γ' ⊢⁰ F → Γ ⊢⁰ F
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halftran≤* h⁺ (neu⁰ x) = neu⁰ (halftran≤⁰ h⁺ x)
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halftran≤* h⁺ (lam h) = lam (halftran≤* ⟨ zero , addhyp⊢⁺⁰ h⁺ ⟩ h)
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halftran≤⁰ h⁺ zero = proj₁ h⁺
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halftran≤⁰ h⁺ (next h) = halftran≤⁰ (proj₂ h⁺) h
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halftran≤⁰ h⁺ (app h h') = app (halftran≤⁰ h⁺ h) (halftran≤* h⁺ h')
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tran≤ : {Γ Γ' Γ'' : Con} → Γ ≤ Γ' → Γ' ≤ Γ'' → Γ ≤ Γ''
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tran≤ {[]} h h' = tt
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tran≤ {x ∷ Γ} h h' = ⟨ halftran≤⁰ h' (proj₁ h) , tran≤ (proj₂ h) h' ⟩
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mon⊩ : {w w' : Con} → w ≤ w' → {x : PV} → w ⊩ x → w' ⊩ x
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mon⊩ h h' = halftran≤⁰ h h'
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⊢*Var : Γ ⊢* Var x → Γ ⊢⁰ Var x
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⊢*Var (neu⁰ x) = x
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UK⁰ : Kripke
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UK⁰ = record {UniversalKripke⁰}
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open Kripke UK⁰
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open UniversalKripke⁰ using (halftran≤⁰)
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-- quote
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⊩ᶠ→⊢ : {F : Form} → {Γ : Con} → Γ ⊩ᶠ F → Γ ⊢* F
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-- unquote
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⊢→⊩ᶠ : {F : Form} → {Γ : Con} → Γ ⊢⁰ F → Γ ⊩ᶠ F
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⊢→⊩ᶠ {Var x} h = h
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⊢→⊩ᶠ {F ⇒ F₁} h {Γ'} iq hF = ⊢→⊩ᶠ {F₁} (app {Γ'} {F} {F₁} (halftran≤⁰ iq h) (⊩ᶠ→⊢ hF))
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⊩ᶠ→⊢ {Var x} h = neu⁰ h
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⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁺⁰ refl⊢⁺⁰) (⊢→⊩ᶠ {F} {F ∷ Γ} zero)))
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--⊩ᶠ→⊢ {F ⇒ G} {Γ} h = lam (⊩ᶠ→⊢ {G} (h (addhyp⊢⁺ refl⊢⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} zero)))
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{-
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⊩ᶠ→⊢ {F} zero = neu⁰ zero
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⊩ᶠ→⊢ {Var x} (next h) = neu⁰ (next {!!})
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⊩ᶠ→⊢ {F ⇒ G} (next h) = neu⁰ (next {!!})
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⊩ᶠ→⊢ {F ⇒ G} (lam h) = lam (⊩ᶠ→⊢ h)
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⊩ᶠ→⊢ {Var x} (app h h₁) = neu⁰ (app {!⊩ᶠ→⊢ h!} (⊩ᶠ→⊢ h₁))
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⊩ᶠ→⊢ {F ⇒ G} (app h h₁) = neu⁰ (app {!!} (⊩ᶠ→⊢ h₁))
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-}
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{-
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⊩ᶠ→⊢ {Var x} zero = neu⁰ zero
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⊩ᶠ→⊢ {Var x} (next h) = neu⁰ (next (⊢*Var (⊩ᶠ→⊢ {Var x} h)))
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⊩ᶠ→⊢ {Var x} (app {A = A} h h₁) = {!!}
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-- neu⁰ (app {A = A} {!!} (⊩ᶠ→⊢ (CompletenessProof.⊢→⊩ᶠ h₁)))
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⊩ᶠ→⊢ {F ⇒ G} {Γ} h = lam (⊩ᶠ→⊢ {G} (h (addhyp⊢⁺ refl⊢⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} zero)))
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-}
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@ -106,15 +106,3 @@ module PropositionalKripke (PV : Set) where
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completeness : {F : Form} → [] ⊫ F → [] ⊢ F
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completeness {F} ⊫F = ⊩ᶠ→⊢ (⊫F tt)
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{- Normalization -}
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norm : [] ⊢ F → [] ⊢ F
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norm x = completeness (⟦ x ⟧)
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-- norm is identity ?!
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idnorm : norm x ≡ x
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idnorm = ?
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-- autonorm : (P₁ P₂ : Prop) → (x₁ : P₁) → (norm x₁ : P₂) → P₁ ≡⊢ P₂
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-- βηnorm : (P₁ P₂ : Prop) → (x₁ : P₁) → (norm x₁ : P₂) → (x₂ : P₂) → norm x₁ ≡ x₂ → P₁ ≡⊢ P₂
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-- autonorm P = {!!}
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--βηnorm P₁ P₂ = ?
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@ -47,29 +47,6 @@ module PropositionalLogic (PV : Set) where
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infix 5 _⊢_
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-- Equality of derivation
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infix 2 _≡⊢_
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data _≡⊢_ : Prop → Prop → Prop where
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refl : Γ ⊢ A ≡⊢ Γ ⊢ A
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{-zero≡⊢ : (A ∷ Γ) ⊢ A ≡⊢ (A' ∷ Γ') ⊢ A'
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next≡⊢ : Γ ⊢ A ≡⊢ Γ' ⊢ A' → (B ∷ Γ) ⊢ A ≡⊢ (B ∷ Γ') ⊢ A'
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lam≡⊢ : (A ∷ Γ) ⊢ B ≡⊢ (A' ∷ Γ') ⊢ B' → Γ ⊢ (A ⇒ B) ≡⊢ Γ' ⊢ (A' ⇒ B')
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app≡⊢ : Γ ⊢ (A ⇒ B) ≡⊢ Γ' ⊢ (A' ⇒ B') → Γ ⊢ A ≡⊢ Γ' ⊢ A' → Γ ⊢ B ≡⊢ Γ' ⊢ B'
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-}
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{-
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-- Reflexivity of equality
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refl≡⊢ : {Γ : Con} → {A : Form} → Γ ⊢ A ≡⊢ Γ ⊢ A
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refl≡⊢ = {!!}
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-- Symmetry of equality
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sym≡⊢ : {Γ Γ' : Con} → {A A' : Form} → Γ ⊢ A ≡⊢ Γ' ⊢ A' → Γ' ⊢ A' ≡⊢ Γ ⊢ A
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sym≡⊢ = {!!}
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-- Transitivity of equality
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tran≡⊢ : {Γ Γ' Γ'' : Con} → {A A' A'' : Form} → Γ ⊢ A ≡⊢ Γ' ⊢ A' → Γ' ⊢ A' ≡⊢ Γ'' ⊢ A'' → Γ ⊢ A ≡⊢ Γ'' ⊢ A''
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tran≡⊢ = {!!}
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-}
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-- Extension of ⊢ to contexts
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_⊢⁺_ : Con → Con → Prop
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Γ ⊢⁺ [] = ⊤
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209
prop.agda
209
prop.agda
@ -7,91 +7,10 @@ open import Data.String.Properties using (_==_)
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open import Data.List using (List; _∷_; [])
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{- Prop -}
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-- ⊥ is a data with no constructor
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-- ⊤ is a record with one always-available constructor
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data ⊥ : Prop where
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record ⊤ : Prop where
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constructor tt
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data _∨_ : Prop → Prop → Prop where
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inj₁ : {P Q : Prop} → P → P ∨ Q
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inj₂ : {P Q : Prop} → Q → P ∨ Q
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record _∧_ (P Q : Prop) : Prop where
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constructor ⟨_,_⟩
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field
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p : P
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q : Q
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infixr 10 _∧_
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infixr 11 _∨_
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-- ∧ elimination
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proj₁ : {P Q : Prop} → P ∧ Q → P
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proj₁ pq = _∧_.p pq
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proj₂ : {P Q : Prop} → P ∧ Q → Q
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proj₂ pq = _∧_.q pq
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-- ¬ is a shorthand for « → ⊥ »
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¬ : Prop → Prop
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¬ P = P → ⊥
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case⊥ : {P : Prop} → ⊥ → P
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case⊥ ()
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-- ∨ elimination
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dis : {P Q S : Prop} → (P ∨ Q) → (P → S) → (Q → S) → S
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dis (inj₁ p) ps qs = ps p
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dis (inj₂ q) ps qs = qs q
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_⇔_ : Prop → Prop → Prop
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P ⇔ Q = (P → Q) ∧ (Q → P)
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data Form : Set where
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Var : String → Form
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_[⇒]_ : Form → Form → Form
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infixr 8 _[⇒]_
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data _≡_ : {A : Set} → A → A → Prop where
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refl : {A : Set} → {x : A} → x ≡ x
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Con = List Form
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variable
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A : Form
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B : Form
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C : Form
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F : Form
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G : Form
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Γ : Con
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Η : Con
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x : String
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y : String
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data _⊢_ : Con → Form → Prop where
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zero : (F ∷ Γ) ⊢ F
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succ : Γ ⊢ F → (G ∷ Γ) ⊢ F
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lam : (F ∷ Γ) ⊢ G → Γ ⊢ (F [⇒] G)
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app : Γ ⊢ (F [⇒] G) → Γ ⊢ F → Γ ⊢ G
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infixr 5 _⊢_
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d-C : [] ⊢ ((Var "Q") [⇒] (Var "R")) [⇒] ((Var "P") [⇒] (Var "Q")) [⇒] (Var "P") [⇒] (Var "R")
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d-C = lam (lam (lam (app (succ (succ zero)) (app (succ zero) zero))))
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Env = String → Prop
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⟦_⟧F : Form → Env → Prop
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⟦ Var x ⟧F ρ = ρ x
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⟦ A [⇒] B ⟧F ρ = (⟦ A ⟧F ρ) → (⟦ B ⟧F ρ)
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⟦_⟧C : Con → Env → Prop
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⟦ [] ⟧C ρ = ⊤
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⟦ A ∷ Γ ⟧C ρ = (⟦ A ⟧F ρ) ∧ (⟦ Γ ⟧C ρ)
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⟦_⟧d : Γ ⊢ F → {ρ : Env} → ⟦ Γ ⟧C ρ → ⟦ F ⟧F ρ
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⟦ zero ⟧d p = proj₁ p
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⟦ succ th ⟧d p = ⟦ th ⟧d (proj₂ p)
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⟦ lam th ⟧d = λ pₐ p₀ → ⟦ th ⟧d ⟨ p₀ , pₐ ⟩
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⟦ app th₁ th₂ ⟧d = λ p → ⟦ th₁ ⟧d p (⟦ th₂ ⟧d p)
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ρ₀ : Env
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ρ₀ "P" = ⊥
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@ -101,46 +20,6 @@ Env = String → Prop
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cex-d : ([] ⊢ (((Var "P") [⇒] (Var "Q")) [⇒] (Var "P"))) → ⊥
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cex-d h = ⟦ h ⟧d {ρ₀} tt λ x → tt
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data ⊢sk : Form → Prop where
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SS : ⊢sk ((A [⇒] B [⇒] C) [⇒] (A [⇒] B) [⇒] A [⇒] C)
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KK : ⊢sk (A [⇒] B [⇒] A)
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app : ⊢sk (A [⇒] B) → ⊢sk A → ⊢sk B
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thm : ([] ⊢ A) ⇔ ⊢sk A
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thm₁ : ⊢sk A → ([] ⊢ A)
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thm₁ SS = lam (lam (lam ( app (app (succ (succ zero)) zero) (app (succ zero) zero))))
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thm₁ KK = lam (lam (succ zero))
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thm₁ (app x x₁) = app (thm₁ x) (thm₁ x₁)
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data _⊢skC_ : Con → Form → Prop where
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zero : (A ∷ Γ) ⊢skC A
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suc : Γ ⊢skC A → (B ∷ Γ) ⊢skC A
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SS : Γ ⊢skC ((A [⇒] B [⇒] C) [⇒] (A [⇒] B) [⇒] A [⇒] C)
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KK : Γ ⊢skC (A [⇒] B [⇒] A)
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app : Γ ⊢skC (A [⇒] B) → Γ ⊢skC A → Γ ⊢skC B
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skC→sk : [] ⊢skC A → ⊢sk A
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skC→sk SS = SS
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skC→sk KK = KK
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skC→sk (app d e) = app (skC→sk d) (skC→sk e)
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lam-sk : (A ∷ Γ) ⊢skC B → Γ ⊢skC (A [⇒] B)
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lam-sk-zero : Γ ⊢skC (A [⇒] A)
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lam-sk-zero {A = A} = app (app SS KK) (KK {B = A})
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lam-sk zero = lam-sk-zero
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lam-sk (suc x) = app KK x
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lam-sk SS = app KK SS
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lam-sk KK = app KK KK
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lam-sk (app x₁ x₂) = app (app SS (lam-sk x₁)) (lam-sk x₂)
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⊢→⊢skC : Γ ⊢ A → Γ ⊢skC A
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⊢→⊢skC zero = zero
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⊢→⊢skC (succ x) = suc (⊢→⊢skC x)
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⊢→⊢skC (lam x) = lam-sk (⊢→⊢skC x)
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⊢→⊢skC (app x x₁) = app (⊢→⊢skC x) (⊢→⊢skC x₁)
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thm = ⟨ (λ x → skC→sk (⊢→⊢skC x)) , thm₁ ⟩
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Pierce = {P Q : Prop} → ((P → Q) → P) → P
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@ -156,35 +35,6 @@ P→TND pierce {P} = pierce {TND P} {⊥} (λ p → case⊥ (nnTND p))
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{- Kripke Models -}
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record Kripke : Set₁ where
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field
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Worlds : Set₀
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_≤_ : Worlds → Worlds → Prop
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refl≤ : {w : Worlds} → w ≤ w
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tran≤ : {a b c : Worlds} → a ≤ b → b ≤ c → a ≤ c
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_⊩_ : Worlds → String → Prop
|
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mon⊩ : {a b : Worlds} → a ≤ b → {p : String} → a ⊩ p → b ⊩ p
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{- Extending ⊩ to Formulas and Contexts -}
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_⊩ᶠ_ : Worlds → Form → Prop
|
||||
w ⊩ᶠ Var x = w ⊩ x
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w ⊩ᶠ (fp [⇒] fq) = {w' : Worlds} → w ≤ w' → w' ⊩ᶠ fp → w' ⊩ᶠ fq
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mon⊩ᶠ : {a b : Worlds} → a ≤ b → a ⊩ᶠ A → b ⊩ᶠ A
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mon⊩ᶠ {Var x} ab aA = mon⊩ ab aA
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mon⊩ᶠ {A [⇒] A₁} ab aA bc cA = aA (tran≤ ab bc) cA
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_⊩ᶜ_ : Worlds → Con → Prop
|
||||
w ⊩ᶜ [] = ⊤
|
||||
w ⊩ᶜ (p ∷ c) = (w ⊩ᶠ p) ∧ (w ⊩ᶜ c)
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mon⊩ᶜ : {a b : Worlds} → a ≤ b → a ⊩ᶜ Γ → b ⊩ᶜ Γ
|
||||
mon⊩ᶜ {[]} ab aΓ = aΓ
|
||||
mon⊩ᶜ {A ∷ Γ} ab aΓ = ⟨ mon⊩ᶠ {A} ab (proj₁ aΓ) , mon⊩ᶜ ab (proj₂ aΓ) ⟩
|
||||
_⊫_ : Con → Form → Prop
|
||||
Γ ⊫ F = {w : Worlds} → w ⊩ᶜ Γ → w ⊩ᶠ F
|
||||
{- Soundness -}
|
||||
⟦_⟧ : Γ ⊢ A → Γ ⊫ A
|
||||
⟦ zero ⟧ = proj₁
|
||||
⟦ succ p ⟧ = λ x → ⟦ p ⟧ (proj₂ x)
|
||||
⟦ lam p ⟧ = λ wΓ w≤ w'A → ⟦ p ⟧ ⟨ w'A , mon⊩ᶜ w≤ wΓ ⟩
|
||||
⟦ app p p₁ ⟧ wΓ = ⟦ p ⟧ wΓ refl≤ (⟦ p₁ ⟧ wΓ)
|
||||
|
||||
{- Pierce is not provable -}
|
||||
|
||||
@ -234,65 +84,6 @@ PierceImpliesw₁ h = Kripke.⟦_⟧ PierceW h {PierceWorld.w₁} tt
|
||||
NotProvable : ¬([] ⊢ FaultyPierce)
|
||||
NotProvable h = Pierce⊥w₁ (PierceImpliesw₁ h)
|
||||
|
||||
{- Universal Kripke -}
|
||||
-- Extension of ⊢ to contexts
|
||||
_⊢⁺_ : Con → Con → Prop
|
||||
Γ ⊢⁺ [] = ⊤
|
||||
Γ ⊢⁺ (F ∷ Γ') = (Γ ⊢ F) ∧ (Γ ⊢⁺ Γ')
|
||||
|
||||
module UniversalKripke where
|
||||
Worlds = Con
|
||||
_≤_ : Con → Con → Prop
|
||||
Γ ≤ Η = Η ⊢⁺ Γ
|
||||
_⊩_ : Con → String → Prop
|
||||
Γ ⊩ x = Γ ⊢ Var x
|
||||
|
||||
data _⊆_ : Con → Con → Prop where
|
||||
zero⊆ : Γ ⊆ Γ
|
||||
next⊆ : Γ ⊆ Η → Γ ⊆ (F ∷ Η)
|
||||
retro⊆ : {Γ Γ' : Con} → {F : Form} → (F ∷ Γ) ⊆ Γ' → Γ ⊆ Γ'
|
||||
retro⊆ {Γ' = []} () -- Impossible to have «F∷Γ ⊆ []»
|
||||
retro⊆ {Γ' = x ∷ Γ'} zero⊆ = next⊆ zero⊆
|
||||
retro⊆ {Γ' = x ∷ Γ'} (next⊆ h) = next⊆ (retro⊆ h)
|
||||
mon⊆≤ : {Γ Γ' : Con} → Γ' ⊆ Γ → Γ ⊢⁺ Γ'
|
||||
mon⊆≤ {[]} zero⊆ = tt
|
||||
mon⊆≤ {x ∷ Γ} zero⊆ = ⟨ zero , mon⊆≤ (next⊆ zero⊆) ⟩
|
||||
mon⊆≤ {x ∷ Γ} {[]} (next⊆ sub) = tt
|
||||
mon⊆≤ {x ∷ Γ} {y ∷ Γ'} (next⊆ sub) = ⟨ succ (proj₁ (mon⊆≤ sub)) , mon⊆≤ (next⊆ (retro⊆ sub)) ⟩
|
||||
|
||||
refl≤ : {Γ : Con} → Γ ⊢⁺ Γ
|
||||
refl≤ = mon⊆≤ zero⊆
|
||||
addhyp : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁺ Γ' → (F ∷ Γ) ⊢⁺ Γ'
|
||||
addhyp {Γ' = []} h = tt
|
||||
addhyp {Γ' = x ∷ Γ'} h = ⟨ succ (proj₁ h) , addhyp (proj₂ h) ⟩
|
||||
halftran≤ : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁺ Γ' → Γ' ⊢ F → Γ ⊢ F
|
||||
halftran≤ h⁺ zero = proj₁ h⁺
|
||||
halftran≤ h⁺ (succ h) = halftran≤ (proj₂ h⁺) h
|
||||
halftran≤ h⁺ (lam h) = lam (halftran≤ ⟨ zero , addhyp h⁺ ⟩ h)
|
||||
halftran≤ h⁺ (app h h') = app (halftran≤ h⁺ h) (halftran≤ h⁺ h')
|
||||
tran≤ : {Γ Γ' Γ'' : Con} → Γ ≤ Γ' → Γ' ≤ Γ'' → Γ ≤ Γ''
|
||||
tran≤ {[]} h h' = tt
|
||||
tran≤ {x ∷ Γ} h h' = ⟨ halftran≤ h' (proj₁ h) , tran≤ (proj₂ h) h' ⟩
|
||||
|
||||
mon⊩ : {w w' : Con} → w ≤ w' → {x : String} → w ⊩ x → w' ⊩ x
|
||||
mon⊩ h h' = halftran≤ h h'
|
||||
|
||||
UK : Kripke
|
||||
UK = record {UniversalKripke}
|
||||
|
||||
module CompletenessProof where
|
||||
open Kripke UK
|
||||
open UniversalKripke using (mon⊆≤ ; zero⊆ ; next⊆ ; halftran≤ ; addhyp)
|
||||
|
||||
⊩ᶠ→⊢ : {F : Form} → {Γ : Con} → Γ ⊩ᶠ F → Γ ⊢ F
|
||||
⊢→⊩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 iq) h) zero)) (⊩ᶠ→⊢ hF))
|
||||
⊩ᶠ→⊢ {Var x} h = h
|
||||
⊩ᶠ→⊢ {F [⇒] F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (mon⊆≤ (next⊆ zero⊆)) (⊢→⊩ᶠ {F} {F ∷ Γ} zero)))
|
||||
|
||||
completeness : {F : Form} → [] ⊫ F → [] ⊢ F
|
||||
completeness {F} ⊫F = ⊩ᶠ→⊢ (⊫F tt)
|
||||
|
||||
|
||||
|
||||
|
||||
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Reference in New Issue
Block a user