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Merge branch 'master' into adding-conjunction

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Incomplete proofs of theorems with \and\and
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Mysaa 2023-05-31 14:13:15 +02:00
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115
ListUtil.agda
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@ -9,11 +9,25 @@ module ListUtil where
T : Set₀
L : List T
L' : List T
L'' : List T
A : T
B : T
-- Definition of sublists
-- Definition of list appartenance
-- The definition uses reflexivity and never any kind of equality
infix 3 _∈_
data _∈_ : T → List T → Prop where
zero∈ : A ∈ A ∷ L
next∈ : A ∈ L → A ∈ B ∷ L
{- RELATIONS BETWEEN LISTS -}
-- We have the following relations
-- ↗ ⊂⁰ ↘
-- ⊆ → ⊂ → ⊂⁺ → ∈*
infix 4 _⊆_ _⊂_ _⊂⁺_ _⊂⁰_ _∈*_
{- ⊆ : We can remove elements but only from the head of the list -}
-- Similar definition : {L L' : List T} → L ⊆ L' ++ L
data _⊆_ : List T → List T → Prop where
zero⊆ : L ⊆ L
@ -25,3 +39,102 @@ module ListUtil where
retro⊆ {L' = B ∷ L'} zero⊆ = next⊆ zero⊆
retro⊆ {L' = B ∷ L'} (next⊆ h) = next⊆ (retro⊆ h)
refl⊆ : L ⊆ L
refl⊆ = zero⊆
tran⊆ : L ⊆ L' → L' ⊆ L'' → L ⊆ L''
tran⊆ zero⊆ h2 = h2
tran⊆ (next⊆ h1) h2 = tran⊆ h1 (retro⊆ h2)
{- ⊂ : We can remove elements anywhere on the list, no duplicates, no reordering -}
data _⊂_ : List T → List T → Prop where
zero⊂ : [] ⊂ L
both⊂ : L ⊂ L' → (A ∷ L) ⊂ (A ∷ L')
next⊂ : L ⊂ L' → L ⊂ (A ∷ L')
⊆→⊂ : L ⊆ L' → L ⊂ L'
refl⊂ : L ⊂ L
⊆→⊂ {L = []} h = zero⊂
⊆→⊂ {L = x ∷ L} zero⊆ = both⊂ (refl⊂)
⊆→⊂ {L = x ∷ L} (next⊆ h) = next⊂ (⊆→⊂ h)
refl⊂ = ⊆→⊂ refl⊆
tran⊂ : L ⊂ L' → L' ⊂ L'' → L ⊂ L''
tran⊂ zero⊂ h2 = zero⊂
tran⊂ (both⊂ h1) (both⊂ h2) = both⊂ (tran⊂ h1 h2)
tran⊂ (both⊂ h1) (next⊂ h2) = next⊂ (tran⊂ (both⊂ h1) h2)
tran⊂ (next⊂ h1) (both⊂ h2) = next⊂ (tran⊂ h1 h2)
tran⊂ (next⊂ h1) (next⊂ h2) = next⊂ (tran⊂ (next⊂ h1) h2)
{- ⊂⁰ : We can remove elements and reorder the list, as long as we don't duplicate the elements -}
-----> We do not have unicity of derivation ([A,A] ⊂⁰ [A,A] can be prove by identity or by swapping its two elements
--> We could do with some counting function, but ... it would not be nice, would it ?
data _⊂⁰_ : List T → List T → Prop where
zero⊂⁰ : _⊂⁰_ {T} [] []
next⊂⁰ : L ⊂⁰ L' → L ⊂⁰ A ∷ L'
both⊂⁰ : L ⊂⁰ L' → A ∷ L ⊂⁰ A ∷ L'
swap⊂⁰ : L ⊂⁰ A ∷ B ∷ L' → L ⊂⁰ B ∷ A ∷ L'
error : L ⊂⁰ L'
-- TODOTODOTODOTODO Fix this definition
{- ⊂⁺ : We can remove and duplicate elements, as long as we don't change the order -}
data _⊂⁺_ : List T → List T → Prop where
zero⊂⁺ : _⊂⁺_ {T} [] []
next⊂⁺ : L ⊂⁺ L' → L ⊂⁺ A ∷ L'
dup⊂⁺ : L ⊂⁺ A ∷ L' → A ∷ L ⊂⁺ A ∷ L'
⊂→⊂⁺ : L ⊂ L' → L ⊂⁺ L'
⊂→⊂⁺ {L' = []} zero⊂ = zero⊂⁺
⊂→⊂⁺ {L' = x ∷ L'} zero⊂ = next⊂⁺ (⊂→⊂⁺ zero⊂)
⊂→⊂⁺ (both⊂ h) = dup⊂⁺ (next⊂⁺ (⊂→⊂⁺ h))
⊂→⊂⁺ (next⊂ h) = next⊂⁺ (⊂→⊂⁺ h)
refl⊂⁺ : L ⊂⁺ L
refl⊂⁺ = ⊂→⊂⁺ refl⊂
tran⊂⁺ : L ⊂⁺ L' → L' ⊂⁺ L'' → L ⊂⁺ L''
tran⊂⁺ zero⊂⁺ zero⊂⁺ = zero⊂⁺
tran⊂⁺ zero⊂⁺ (next⊂⁺ h2) = next⊂⁺ h2
tran⊂⁺ (next⊂⁺ h1) (next⊂⁺ h2) = next⊂⁺ (tran⊂⁺ (next⊂⁺ h1) h2)
tran⊂⁺ (next⊂⁺ h1) (dup⊂⁺ h2) = tran⊂⁺ h1 h2
tran⊂⁺ (dup⊂⁺ h1) (next⊂⁺ h2) = next⊂⁺ (tran⊂⁺ (dup⊂⁺ h1) h2)
tran⊂⁺ (dup⊂⁺ h1) (dup⊂⁺ h2) = dup⊂⁺ (tran⊂⁺ h1 (dup⊂⁺ h2))
retro⊂⁺ : A ∷ L ⊂⁺ L' → L ⊂⁺ L'
retro⊂⁺ (next⊂⁺ h) = next⊂⁺ (retro⊂⁺ h)
retro⊂⁺ (dup⊂⁺ h) = h
{- ∈* : We can remove or duplicate elements and we can change their order -}
-- The weakest of all relations on lists
-- L ∈* L' if all elements of L exists in L' (no consideration for order nor duplication)
data _∈*_ : List T → List T → Prop where
zero∈* : [] ∈* L
next∈* : A ∈ L → L' ∈* L → (A ∷ L') ∈* L
-- Founding principle
mon∈∈* : A ∈ L → L ∈* L' → A ∈ L'
mon∈∈* zero∈ (next∈* x hl) = x
mon∈∈* (next∈ ha) (next∈* x hl) = mon∈∈* ha hl
-- We show that the relation is reflexive and is implied by ⊆
refl∈* : L ∈* L
⊂⁺→∈* : L ⊂⁺ L' → L ∈* L'
refl∈* {L = []} = zero∈*
refl∈* {L = x ∷ L} = next∈* zero∈ (⊂⁺→∈* (next⊂⁺ refl⊂⁺))
⊂⁺→∈* zero⊂⁺ = refl∈*
⊂⁺→∈* {L = []} (next⊂⁺ h) = zero∈*
⊂⁺→∈* {L = x ∷ L} (next⊂⁺ h) = next∈* (next∈ (mon∈∈* zero∈ (⊂⁺→∈* h))) (⊂⁺→∈* (retro⊂⁺ (next⊂⁺ h)))
⊂⁺→∈* (dup⊂⁺ h) = next∈* zero∈ (⊂⁺→∈* h)
-- Allows to grow ∈* to the right
right∈* : L ∈* L' → L ∈* (A ∷ L')
right∈* zero∈* = zero∈*
right∈* (next∈* x h) = next∈* (next∈ x) (right∈* h)
both∈* : L ∈* L' → (A ∷ L) ∈* (A ∷ L')
both∈* zero∈* = next∈* zero∈ zero∈*
both∈* (next∈* x h) = next∈* zero∈ (next∈* (next∈ x) (right∈* h))
tran∈* : L ∈* L' → L' ∈* L'' → L ∈* L''
tran∈* {L = []} = λ x x₁ → zero∈*
tran∈* {L = x ∷ L} (next∈* x₁ h1) h2 = next∈* (mon∈∈* x₁ h2) (tran∈* h1 h2)
⊆→∈* : L ⊆ L' → L ∈* L'
⊆→∈* h = ⊂⁺→∈* (⊂→⊂⁺ (⊆→⊂ h))

46
Prop.agda
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@ -1,46 +0,0 @@
{-# OPTIONS --prop #-}
module Prop where
-- ⊥ is a data with no constructor
-- ⊤ is a record with one always-available constructor
data ⊥ : Prop where
record ⊤ : Prop where
constructor tt
data _∨_ : Prop → Prop → Prop where
inj₁ : {P Q : Prop} → P → P ∨ Q
inj₂ : {P Q : Prop} → Q → P ∨ Q
record _∧_ (P Q : Prop) : Prop where
constructor ⟨_,_⟩
field
p : P
q : Q
infixr 10 _∧_
infixr 11 _∨_
-- ∧ elimination
proj₁ : {P Q : Prop} → P ∧ Q → P
proj₁ pq = _∧_.p pq
proj₂ : {P Q : Prop} → P ∧ Q → Q
proj₂ pq = _∧_.q pq
-- ¬ is a shorthand for « → ⊥ »
¬ : Prop → Prop
¬ P = P → ⊥
-- ⊥ elimination
case⊥ : {P : Prop} → ⊥ → P
case⊥ ()
-- ∨ elimination
dis : {P Q S : Prop} → (P ∨ Q) → (P → S) → (Q → S) → S
dis (inj₁ p) ps qs = ps p
dis (inj₂ q) ps qs = qs q
-- ⇔ shorthand
_⇔_ : Prop → Prop → Prop
P ⇔ Q = (P → Q) ∧ (Q → P)

6
PropUtil.agda
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@ -44,3 +44,9 @@ module PropUtil where
-- ⇔ shorthand
_⇔_ : Prop → Prop → Prop
P ⇔ Q = (P → Q) ∧ (Q → P)
-- Syntactic sugar for writing applications
infixr 200 _$_
_$_ : {T U : Prop} → (T → U) → T → U
h $ t = h t

63
PropositionalKripke.agda
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@ -60,11 +60,13 @@ module PropositionalKripke (PV : Set) where
{- Soundness -}
⟦_⟧ : Γ ⊢ F → Γ ⊫ F
⟦ zero ⟧ = proj₁
⟦ next p ⟧ = λ x → ⟦ p ⟧ (proj₂ x)
⟦ zero zero∈ ⟧ wΓ = proj₁ wΓ
⟦ 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 ⟧ = {!!}
@ -83,15 +85,15 @@ module PropositionalKripke (PV : Set) where
mon⊩ = λ ba bx → halftran⊢⁺ ba bx
}
open Kripke UK
-- Now we can prove that ⊩ᶠ and ⊢ act in the same way
⊩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 {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→⊢ {Var x} h = h
⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁺ refl⊢⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} zero)))
⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁺ (right∈* refl∈*) refl⊢⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} (zero zero∈))))
⊩ᶠ→⊢ {F ∧∧ G} ⟨ hF , hG ⟩ = {!!}
-- Finally, we can deduce completeness of the Kripke model
@ -99,7 +101,7 @@ module PropositionalKripke (PV : Set) where
completeness {F} ⊫F = ⊩ᶠ→⊢ (⊫F tt)
module NormalizationProof where
-- First we define the Universal model with (⊢⁰⁺)⁻¹ as world order
-- It is slightly different from the last Model, but proofs are the same
UK⁰ : Kripke
@ -122,5 +124,50 @@ 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→⊢ {Var x} h = neu⁰ h
⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁰⁺ refl⊢⁰⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} zero)))
⊩ᶠ→⊢ {F ⇒ F₁} {Γ} h = lam (⊩ᶠ→⊢ (h (addhyp⊢⁰⁺ (right∈* refl∈*) refl⊢⁰⁺) (⊢→⊩ᶠ {F} {F ∷ Γ} (zero zero∈))))
⊩ᶠ→⊢ {F ∧∧ G} h = {!!}
module OtherProofs where
-- We will try to define the Kripke models using the following embeddings
-- Strongest is using the ⊢⁺ relation directly
-- -> See module CompletenessProof
-- We can also use the relation ⊢⁰⁺, which is compatible with ⊢⁰ and ⊢*
-- -> See module NormalizationProof
{- Renamings : ∈* -}
UK∈* : Kripke
UK∈* = record {
Worlds = Con;
_≤_ = _∈*_;
refl≤ = refl∈*;
tran≤ = tran∈*;
_⊩_ = λ Γ x → Γ ⊢ Var x;
mon⊩ = λ x x₁ → addhyp⊢ x x₁
}
{-
{- Weakening anywhere, order preserving, duplication authorized : ⊂⁺ -}
UK⊂⁺ : Kripke
UK⊂⁺ = record {
Worlds = Con;
_≤_ = _⊂⁺_;
refl≤ = refl⊂⁺;
tran≤ = tran⊂⁺;
_⊩_ = λ Γ x → Γ ⊢ Var x;
mon⊩ = λ x x₁ → addhyp⊢ x x₁
}
-}
{- Weakening anywhere, no duplication, order preserving : ⊂ -}
{- Weakening at the end : ⊆-}
-- This is exactly our relation ⊆

157
PropositionalKripkeGeneral.agda Normal file
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@ -0,0 +1,157 @@
{-# OPTIONS --prop #-}
module PropositionalKripkeGeneral (PV : Set) where
open import ListUtil
open import PropUtil
open import PropositionalLogic PV using (Form; Var; _⇒_; Con)
open import PropositionalKripke PV using (Kripke)
record Preorder (T : Set₀) : Set₁ where
constructor order
field
_≤_ : T → T → Prop
refl≤ : {a : T} → a ≤ a
tran≤ : {a b c : T} → a ≤ b → b ≤ c → a ≤ c
[_]ᵒᵖ : {T : Set₀} → Preorder T → Preorder T
[_]ᵒᵖ o = order (λ a b → (Preorder._≤_ o) b a) (Preorder.refl≤ o) (λ h₁ h₂ → (Preorder.tran≤ o) h₂ h₁)
record NormalAndNeutral : Set₁ where
field
_⊢⁰_ : Con → Form → Prop
_⊢*_ : 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
neu⁰ : {Γ : Con} → {x : PV} → Γ ⊢⁰ Var x → Γ ⊢* Var x
lam : {Γ : Con} → {F G : Form} → (F ∷ Γ) ⊢* G → Γ ⊢* (F ⇒ G)
record NormalizationFrame : Set₁ where
field
o : Preorder Con
nn : NormalAndNeutral
retro : {Γ Δ : Con} → {F : Form} → (Preorder._≤_ o) Γ Δ → (Preorder._≤_ o) Γ (F ∷ Δ)
⊢tran : {Γ Δ : Con} → {F : Form} → (Preorder._≤_ o) Γ Δ → (NormalAndNeutral._⊢⁰_ nn) Γ F → (NormalAndNeutral._⊢⁰_ nn) Δ F
open Preorder o
open NormalAndNeutral nn
all : Con → PV → Prop
all Γ x = ⊤
UK : Kripke
UK = record {
Worlds = Con;
_≤_ = _≤_;
refl≤ = refl≤;
tran≤ = tran≤;
_⊩_ = λ Γ x → Γ ⊢⁰ Var x;
mon⊩ = λ Γ h → ⊢tran Γ h
}
open Kripke UK
-- q is quote, u is unquote
q : {F : Form} → {Γ : Con} → Γ ⊩ᶠ F → Γ ⊢* F
u : {F : Form} → {Γ : Con} → Γ ⊢⁰ F → Γ ⊩ᶠ F
u {Var x} h = h
u {F ⇒ F₁} h {Γ'} iq hF = u {F₁} (app {Γ'} {F} {F₁} (⊢tran iq h) (q hF))
q {Var x} h = neu⁰ h
q {F ⇒ F₁} {Γ} h = lam (q (h (retro (Preorder.refl≤ o)) (u {F} {F ∷ Γ} zero)))
module NormalizationTests where
{- Now using our records -}
open import PropositionalLogic PV hiding (Form; Var; _⇒_; Con)
ClassicNN : NormalAndNeutral
ClassicNN = record
{
_⊢⁰_ = _⊢⁰_ ;
_⊢*_ = _⊢*_ ;
zero = zero zero∈ ;
app = app ;
neu⁰ = neu⁰ ;
lam = lam
}
BiggestNN : NormalAndNeutral
BiggestNN = record
{
_⊢⁰_ = _⊢_ ;
_⊢*_ = _⊢_ ;
zero = zero zero∈ ;
app = app ;
neu⁰ = λ x → x ;
lam = lam
}
PO⊢⁺ = [ order {Con} _⊢⁺_ refl⊢⁺ tran⊢⁺ ]ᵒᵖ
PO⊢⁰⁺ = [ order {Con} _⊢⁰⁺_ refl⊢⁰⁺ tran⊢⁰⁺ ]ᵒᵖ
PO∈* = order {Con} _∈*_ refl∈* tran∈*
PO⊂⁺ = order {Con} _⊂⁺_ refl⊂⁺ tran⊂⁺
PO⊂ = order {Con} _⊂_ refl⊂ tran⊂
PO⊆ = order {Con} _⊆_ refl⊆ tran⊆
-- Completeness Proofs
Frame⊢ : NormalizationFrame
Frame⊢ = record
{
o = PO⊢⁺ ;
nn = BiggestNN ;
retro = λ s → addhyp⊢⁺ (right∈* refl∈*) s ;
⊢tran = halftran⊢⁺
}
Frame⊢⁰ : NormalizationFrame
Frame⊢⁰ = record
{
o = PO⊢⁰⁺ ;
nn = ClassicNN ;
retro = λ s → addhyp⊢⁰⁺ (right∈* refl∈*) s ;
⊢tran = halftran⊢⁰⁺⁰
}
Frame∈* : NormalizationFrame
Frame∈* = record
{
o = PO∈* ;
nn = ClassicNN ;
retro = right∈* ;
⊢tran = λ s h → halftran⊢⁰⁺⁰ (mon∈*⊢⁰⁺ s) h
}
Frame⊂⁺ : NormalizationFrame
Frame⊂⁺ = record
{
o = PO⊂⁺ ;
nn = ClassicNN ;
retro = next⊂⁺ ;
⊢tran = λ s h → halftran⊢⁰⁺⁰ (mon∈*⊢⁰⁺ $ ⊂⁺→∈* s) h
}
Frame⊂ : NormalizationFrame
Frame⊂ = record
{
o = PO⊂ ;
nn = ClassicNN ;
retro = next⊂ ;
⊢tran = λ s h → halftran⊢⁰⁺⁰ (mon∈*⊢⁰⁺ $ ⊂⁺→∈* $ ⊂→⊂⁺ s) h
}
Frame⊆ : NormalizationFrame
Frame⊆ = record
{
o = PO⊆ ;
nn = ClassicNN ;
retro = next⊆ ;
⊢tran = λ s h → halftran⊢⁰⁺⁰ (mon∈*⊢⁰⁺ $ ⊂⁺→∈* $ ⊂→⊂⁺ $ ⊆→⊂ s) h
}

112
PropositionalLogic.agda
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@ -27,6 +27,7 @@ module PropositionalLogic (PV : Set) where
G : Form
Γ : Con
Γ' : Con
Γ'' : Con
x : PV
@ -34,8 +35,7 @@ module PropositionalLogic (PV : Set) where
{--- DEFINITION OF ⊢ ---}
data _⊢_ : Con → Form → Prop where
zero : (A ∷ Γ) ⊢ A
next : Γ ⊢ A → (B ∷ Γ) ⊢ A
zero : A ∈ Γ → Γ ⊢ A
lam : (A ∷ Γ) ⊢ B → Γ ⊢ (A ⇒ B)
app : Γ ⊢ (A ⇒ B) → Γ ⊢ A → Γ ⊢ B
andi : Γ ⊢ A → Γ ⊢ B → Γ ⊢ A ∧∧ B
@ -43,6 +43,20 @@ module PropositionalLogic (PV : Set) where
ande₂ : Γ ⊢ A ∧∧ B → Γ ⊢ B
infix 5 _⊢_
zero⊢ : (A ∷ Γ) ⊢ A
zero⊢ = zero zero∈
one⊢ : (B ∷ A ∷ Γ) ⊢ A
one⊢ = zero (next∈ zero∈)
-- We can add hypotheses to a proof
addhyp⊢ : Γ ∈* Γ' → Γ ⊢ A → Γ' ⊢ A
addhyp⊢ s (zero x) = zero (mon∈∈* x s)
addhyp⊢ s (lam h) = lam (addhyp⊢ (both∈* s) h)
addhyp⊢ s (app h h₁) = app (addhyp⊢ s h) (addhyp⊢ s h₁)
addhyp⊢ s (andi h₁ h₂) = andi (addhyp⊢ s h₁) (addhyp⊢ s h₂)
addhyp⊢ s (ande₁ h) = ande₁ (addhyp⊢ s h)
addhyp⊢ s (ande₂ h) = ande₂ (addhyp⊢ s h)
-- Extension of ⊢ to contexts
_⊢⁺_ : Con → Con → Prop
@ -50,28 +64,32 @@ module PropositionalLogic (PV : Set) where
Γ ⊢⁺ (F ∷ Γ') = (Γ ⊢ F) ∧ (Γ ⊢⁺ Γ')
infix 5 _⊢⁺_
-- We show that the relation respects ∈*
mon∈*⊢⁺ : Γ' ∈* Γ → Γ ⊢⁺ Γ'
mon∈*⊢⁺ zero∈* = tt
mon∈*⊢⁺ (next∈* x h) = ⟨ (zero x) , (mon∈*⊢⁺ h) ⟩
-- The relation respects ⊆
mon⊆⊢⁺ : Γ' ⊆ Γ → Γ ⊢⁺ Γ'
mon⊆⊢⁺ {[]} sub = tt
mon⊆⊢⁺ {x ∷ Γ} zero⊆ = ⟨ zero , mon⊆⊢⁺ (next⊆ zero⊆) ⟩
mon⊆⊢⁺ {x ∷ Γ} (next⊆ sub) = ⟨ (next (proj₁ (mon⊆⊢⁺ sub)) ) , mon⊆⊢⁺ (next⊆ (retro⊆ sub)) ⟩
mon⊆⊢⁺ h = mon∈*⊢⁺ (⊆→∈* h)
-- The relation is reflexive
refl⊢⁺ : Γ ⊢⁺ Γ
refl⊢⁺ {[]} = tt
refl⊢⁺ {x ∷ Γ} = ⟨ zero , mon⊆⊢⁺ (next⊆ zero⊆) ⟩
-- We can add hypotheses to the left
addhyp⊢⁺ : Γ ⊢⁺ Γ' → (A ∷ Γ) ⊢⁺ Γ'
addhyp⊢⁺ {Γ' = []} h = tt
addhyp⊢⁺ {Γ' = A ∷ Γ'} h = ⟨ next (proj₁ h) , addhyp⊢⁺ (proj₂ h) ⟩
refl⊢⁺ {x ∷ Γ} = ⟨ zero⊢ , mon⊆⊢⁺ (next⊆ zero⊆) ⟩
-- We can add hypotheses to to a proof
addhyp⊢⁺ : Γ ∈* Γ' → Γ ⊢⁺ Γ'' → Γ' ⊢⁺ Γ''
addhyp⊢⁺ {Γ'' = []} s h = tt
addhyp⊢⁺ {Γ'' = x ∷ Γ''} s ⟨ Γx , ΓΓ'' ⟩ = ⟨ addhyp⊢ s Γx , addhyp⊢⁺ s ΓΓ'' ⟩
-- The relation respects ⊢
halftran⊢⁺ : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁺ Γ' → Γ' ⊢ F → Γ ⊢ F
halftran⊢⁺ h⁺ zero = proj₁ h⁺
halftran⊢⁺ h⁺ (next 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')
halftran⊢⁺ {Γ' = F ∷ Γ'} h⁺ (zero zero∈) = proj₁ h⁺
halftran⊢⁺ {Γ' = F ∷ Γ'} h⁺ (zero (next∈ x)) = halftran⊢⁺ (proj₂ h⁺) (zero x)
halftran⊢⁺ h⁺ (lam h) = lam (halftran⊢⁺ ⟨ (zero zero∈) , (addhyp⊢⁺ (right∈* refl∈*) h⁺) ⟩ h)
halftran⊢⁺ h⁺ (app h h₁) = app (halftran⊢⁺ h⁺ h) (halftran⊢⁺ h⁺ h₁)
halftran⊢⁺ h⁺ (andi hf hg) = andi (halftran⊢⁺ h⁺ hf) (halftran⊢⁺ h⁺ hg)
halftran⊢⁺ h⁺ (ande₁ hfg) = ande₁ (halftran⊢⁺ h⁺ hfg)
halftran⊢⁺ h⁺ (ande₂ hfg) = ande₂ (halftran⊢⁺ h⁺ hfg)
@ -89,8 +107,7 @@ module PropositionalLogic (PV : Set) where
-- ⊢* are normal forms
mutual
data _⊢⁰_ : Con → Form → Prop where
zero : (A ∷ Γ) ⊢⁰ A
next : Γ ⊢⁰ A → (B ∷ Γ) ⊢⁰ A
zero : A ∈ Γ → Γ ⊢⁰ A
app : Γ ⊢⁰ (A ⇒ B) → Γ ⊢* A → Γ ⊢⁰ B
data _⊢*_ : Con → Form → Prop where
neu⁰ : Γ ⊢⁰ Var x → Γ ⊢* Var x
@ -101,42 +118,53 @@ module PropositionalLogic (PV : Set) where
-- Both are tighter than ⊢
⊢⁰→⊢ : Γ ⊢⁰ F → Γ ⊢ F
⊢*→⊢ : Γ ⊢* F → Γ ⊢ F
⊢⁰→⊢ zero = zero
⊢⁰→⊢ (next h) = next (⊢⁰→⊢ h)
⊢⁰→⊢ (zero h) = zero h
⊢⁰→⊢ (app h x) = app (⊢⁰→⊢ h) (⊢*→⊢ x)
⊢*→⊢ (neu⁰ x) = ⊢⁰→⊢ x
⊢*→⊢ (lam h) = lam (⊢*→⊢ h)
-- 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⊢* s (neu⁰ x) = neu⁰ (addhyp⊢⁰ s x)
addhyp⊢* s (lam h) = lam (addhyp⊢* (both∈* s) h)
-- Extension of ⊢⁰ to contexts
-- i.e. there is a neutral proof for any element
_⊢⁰⁺_ : Con → Con → Prop
Γ ⊢⁰⁺ [] = ⊤
Γ ⊢⁰⁺ (F ∷ Γ') = (Γ ⊢⁰ F) ∧ (Γ ⊢⁰⁺ Γ')
infix 5 _⊢⁰⁺_
-- The relation respects ∈*
mon∈*⊢⁰⁺ : Γ' ∈* Γ → Γ ⊢⁰⁺ Γ'
mon∈*⊢⁰⁺ zero∈* = tt
mon∈*⊢⁰⁺ (next∈* x h) = ⟨ (zero x) , (mon∈*⊢⁰⁺ h) ⟩
-- The relation respects ⊆
mon⊆⊢⁰⁺ : Γ' ⊆ Γ → Γ ⊢⁰⁺ Γ'
mon⊆⊢⁰⁺ {[]} sub = tt
mon⊆⊢⁰⁺ {x ∷ Γ} zero⊆ = ⟨ zero , mon⊆⊢⁰⁺ (next⊆ zero⊆) ⟩
mon⊆⊢⁰⁺ {x ∷ Γ} (next⊆ sub) = ⟨ (next (proj₁ (mon⊆⊢⁰⁺ sub)) ) , mon⊆⊢⁰⁺ (next⊆ (retro⊆ sub)) ⟩
mon⊆⊢⁰⁺ h = mon∈*⊢⁰⁺ (⊆→∈* h)
-- This relation is reflexive
refl⊢⁰⁺ : Γ ⊢⁰⁺ Γ
refl⊢⁰⁺ {[]} = tt
refl⊢⁰⁺ {x ∷ Γ} = ⟨ zero , mon⊆⊢⁰⁺ (next⊆ zero⊆) ⟩
refl⊢⁰⁺ {x ∷ Γ} = ⟨ zero zero∈ , mon⊆⊢⁰⁺ (next⊆ zero⊆) ⟩
-- A useful lemma, that we can add hypotheses
addhyp⊢⁰⁺ : Γ ⊢⁰⁺ Γ' → (A ∷ Γ) ⊢⁰⁺ Γ'
addhyp⊢⁰⁺ {Γ' = []} h = tt
addhyp⊢⁰⁺ {Γ' = A ∷ Γ'} h = ⟨ next (proj₁ h) , addhyp⊢⁰⁺ (proj₂ h) ⟩
addhyp⊢⁰⁺ : Γ ∈* Γ' → Γ ⊢⁰⁺ Γ'' → Γ' ⊢⁰⁺ Γ''
addhyp⊢⁰⁺ {Γ'' = []} s h = tt
addhyp⊢⁰⁺ {Γ'' = A ∷ Γ'} s ⟨ Γx , ΓΓ'' ⟩ = ⟨ addhyp⊢⁰ s Γx , addhyp⊢⁰⁺ s ΓΓ'' ⟩
-- The relation preserves ⊢⁰ and ⊢*
halftran⊢⁰⁺* : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁰⁺ Γ' → Γ' ⊢* F → Γ ⊢* F
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 , addhyp⊢⁰⁺ h⁺ ⟩ h)
halftran⊢⁰⁺⁰ h⁺ zero = proj₁ h⁺
halftran⊢⁰⁺⁰ h⁺ (next h) = halftran⊢⁰⁺⁰ (proj₂ h⁺) h
halftran⊢⁰⁺* h⁺ (lam h) = lam (halftran⊢⁰⁺* ⟨ zero zero∈ , addhyp⊢⁰⁺ (right∈* refl∈*) h⁺ ⟩ h)
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')
-- The relation is transitive
@ -161,8 +189,8 @@ module PropositionalLogic (PV : Set) where
⟦ A ∷ Γ ⟧ᶜ ρ = (⟦ A ⟧ᶠ ρ) ∧ (⟦ Γ ⟧ᶜ ρ)
⟦_⟧ᵈ : Γ ⊢ A → {ρ : Env} → ⟦ Γ ⟧ᶜ ρ → ⟦ A ⟧ᶠ ρ
⟦ zero ⟧ᵈ p = proj₁ p
⟦ next th ⟧ᵈ p = ⟦ th ⟧ᵈ (proj₂ p)
⟦_⟧ᵈ {x ∷ Γ} (zero zero∈) p = proj₁ p
⟦_⟧ᵈ {x ∷ Γ} (zero (next∈ h)) p = ⟦ zero h ⟧ᵈ (proj₂ p)
⟦ lam th ⟧ᵈ = λ pₐ p₀ → ⟦ th ⟧ᵈ ⟨ p₀ , pₐ ⟩
⟦ app th₁ th₂ ⟧ᵈ = λ p → ⟦ th₁ ⟧ᵈ p (⟦ th₂ ⟧ᵈ p)
⟦ andi hf hg ⟧ᵈ = λ p → ⟨ ⟦ hf ⟧ᵈ p , ⟦ hg ⟧ᵈ p ⟩
@ -184,8 +212,7 @@ module PropositionalLogic (PV : Set) where
app : ⊢sk (A ⇒ B) → ⊢sk A → ⊢sk B
data _⊢skC_ : Con → Form → Prop where
zero : (A ∷ Γ) ⊢skC A
next : Γ ⊢skC A → (B ∷ Γ) ⊢skC A
zero : A ∈ Γ → Γ ⊢skC A
SS : Γ ⊢skC ((A ⇒ B ⇒ C) ⇒ (A ⇒ B) ⇒ A ⇒ C)
KK : Γ ⊢skC (A ⇒ B ⇒ A)
ANDi : Γ ⊢skC (A ⇒ B ⇒ (A ∧∧ B))
@ -197,11 +224,11 @@ module PropositionalLogic (PV : Set) where
⊢⇔⊢sk : ([] ⊢ A) ⇔ ⊢sk A
⊢sk→⊢ : ⊢sk A → ([] ⊢ A)
⊢sk→⊢ SS = lam (lam (lam ( app (app (next (next zero)) zero) (app (next zero) zero))))
⊢sk→⊢ KK = lam (lam (next zero))
⊢sk→⊢ ANDi = lam (lam (andi (next zero) zero))
⊢sk→⊢ ANDe₁ = lam (ande₁ zero)
⊢sk→⊢ ANDe₂ = lam (ande₂ zero)
⊢sk→⊢ SS = lam (lam (lam ( app (app (zero $ next∈ $ next∈ zero∈) (zero zero∈)) (app (zero $ next∈ $ zero∈) (zero zero∈)))))
⊢sk→⊢ KK = lam (lam (zero $ next∈ $ zero∈))
⊢sk→⊢ ANDi = lam (lam (andi (zero $ next∈ $ zero∈) (zero zero∈)))
⊢sk→⊢ ANDe₁ = lam (ande₁ (zero zero∈))
⊢sk→⊢ ANDe₂ = lam (ande₂ (zero zero∈))
⊢sk→⊢ (app x x₁) = app (⊢sk→⊢ x) (⊢sk→⊢ x₁)
skC→sk : [] ⊢skC A → ⊢sk A
@ -216,8 +243,8 @@ module PropositionalLogic (PV : Set) where
lam-sk : (A ∷ Γ) ⊢skC B → Γ ⊢skC (A ⇒ B)
lam-sk-zero : Γ ⊢skC (A ⇒ A)
lam-sk-zero {A = A} = app (app SS KK) (KK {B = A})
lam-sk zero = lam-sk-zero
lam-sk (next x) = app KK x
lam-sk (zero zero∈) = lam-sk-zero
lam-sk (zero (next∈ h)) = app KK (zero h)
lam-sk SS = app KK SS
lam-sk KK = app KK KK
lam-sk ANDi = app KK (app (app SS (app (app SS (app KK SS)) (app (app SS (app KK KK)) (app (app SS (app KK ANDi)) (lam-sk-zero))))) (app KK lam-sk-zero))
@ -227,8 +254,7 @@ module PropositionalLogic (PV : Set) where
⊢→⊢skC : Γ ⊢ A → Γ ⊢skC A
⊢→⊢skC zero = zero
⊢→⊢skC (next x) = next (⊢→⊢skC x)
⊢→⊢skC (zero h) = zero h
⊢→⊢skC (lam x) = lam-sk (⊢→⊢skC x)
⊢→⊢skC (app x x₁) = app (⊢→⊢skC x) (⊢→⊢skC x₁)
⊢→⊢skC (andi x y) = app (app ANDi (⊢→⊢skC x)) (⊢→⊢skC y)

44
Readme.agda
View File

@ -12,7 +12,7 @@ open import PropositionalLogic String
-- Here is an example of a propositional formula and its proof
-- The formula is (Q → R) → (P → Q) → P → R
d-C : [] ⊢ ((Var "Q") ⇒ (Var "R")) ⇒ ((Var "P") ⇒ (Var "Q")) ⇒ (Var "P") ⇒ (Var "R")
d-C = lam (lam (lam (app (next (next zero)) (app (next zero) zero))))
d-C = lam (lam (lam (app (zero $ next∈ $ next∈ zero∈) (app (zero $ next∈ zero∈) (zero zero∈)))))
-- We can with the basic interpretation ⟦_⟧ prove that some formulæ are not provable
-- For example, we can disprove (P → Q) → P 's provability as we can construct an
@ -115,15 +115,39 @@ module PierceDisproof where
PierceNotProvable h = Pierce⊥w₁ (⟦ h ⟧ {w₁} tt)
module CompletenessAndNormalization where
-- With Kripke models, we can even prove completeness
-- Using the Universal Kripke Model
completenessQuote = CompletenessProof.⊩ᶠ→⊢
completenessUnquote = CompletenessProof.⊢→⊩ᶠ
-- With a slightly different universal model (using normal and neutral forms),
-- we can make a normalization proof
normalizationQuote = NormalizationProof.⊩ᶠ→⊢
normalizationUnquote = NormalizationProof.⊢→⊩ᶠ
-- This normalization proof has been made in the biggest Kripke model possible
-- that is, the one using the relation ⊢⁰⁺
-- But we can also prove it with tighter relations: ∈*, ⊂⁺, ⊂, ⊆
-- As all those proofs are really similar, we created a NormalizationFrame structure
-- that computes most of the proof with only a few lemmas
open import PropositionalKripkeGeneral String
-- We now have access to quote and unquote functions with this
u1 = NormalizationFrame.u NormalizationTests.Frame⊢
q1 = NormalizationFrame.q NormalizationTests.Frame⊢
u2 = NormalizationFrame.u NormalizationTests.Frame⊢⁰
q2 = NormalizationFrame.q NormalizationTests.Frame⊢⁰
u3 = NormalizationFrame.u NormalizationTests.Frame∈*
q3 = NormalizationFrame.q NormalizationTests.Frame∈*
u4 = NormalizationFrame.u NormalizationTests.Frame⊂⁺
q4 = NormalizationFrame.q NormalizationTests.Frame⊂⁺
u5 = NormalizationFrame.u NormalizationTests.Frame⊂
q5 = NormalizationFrame.q NormalizationTests.Frame⊂
u6 = NormalizationFrame.u NormalizationTests.Frame⊆
q6 = NormalizationFrame.q NormalizationTests.Frame⊆