From 1dc1826f496ede73f170cbce3928362cd5ab4abc Mon Sep 17 00:00:00 2001
From: Mysaa <samy.avrillon@ens-lyon.fr>
Date: Thu, 20 Jul 2023 11:55:25 +0200
Subject: [PATCH] Changes files names, updated Readme

---
 FinitaryFirstOrderLogic.agda => FFOL.agda     |   4 +-
 FFOLCompleteness.agda                         |   4 +-
 FFOLInitial.agda                              |   2 +-
 InfinitaryFirstOrderLogic.agda => IFOL.agda   |   9 +-
 ...ryFirstOrderKripke.agda => IFOLKripke.agda |   9 +-
 ...ipkeGeneral.agda => IFOLNormalization.agda |  15 +-
 Readme.agda                                   |  50 +++++--
 PropositionalLogic.agda => ZOL.agda           |   4 +-
 PropositionalKripke.agda => ZOLKripke.agda    |   6 +-
 ...ripkeGeneral.agda => ZOLNormalization.agda |  14 +-
 fol-subst.agda                                | 138 ------------------
 fol.agda                                      | 124 ----------------
 12 files changed, 66 insertions(+), 313 deletions(-)
 rename FinitaryFirstOrderLogic.agda => FFOL.agda (99%)
 rename InfinitaryFirstOrderLogic.agda => IFOL.agda (97%)
 rename InfinitaryFirstOrderKripke.agda => IFOLKripke.agda (93%)
 rename InfinitaryFirstOrderKripkeGeneral.agda => IFOLNormalization.agda (92%)
 rename PropositionalLogic.agda => ZOL.agda (99%)
 rename PropositionalKripke.agda => ZOLKripke.agda (95%)
 rename PropositionalKripkeGeneral.agda => ZOLNormalization.agda (93%)
 delete mode 100644 fol-subst.agda
 delete mode 100644 fol.agda

diff --git a/FinitaryFirstOrderLogic.agda b/FFOL.agda
similarity index 99%
rename from FinitaryFirstOrderLogic.agda
rename to FFOL.agda
index 245a615..9e81cea 100644
--- a/FinitaryFirstOrderLogic.agda
+++ b/FFOL.agda
@@ -1,8 +1,8 @@
 {-# OPTIONS --prop --rewriting #-}
 
 open import PropUtil
-
-module FinitaryFirstOrderLogic where
+ 
+module FFOL where
 
   open import Agda.Primitive
   open import ListUtil
diff --git a/FFOLCompleteness.agda b/FFOLCompleteness.agda
index a1caac2..e122dff 100644
--- a/FFOLCompleteness.agda
+++ b/FFOLCompleteness.agda
@@ -5,7 +5,7 @@ open import PropUtil
 module FFOLCompleteness where
 
   open import Agda.Primitive
-  open import FinitaryFirstOrderLogic
+  open import FFOL
   open import ListUtil
 
   record Family : Set (lsuc (ℓ¹)) where
@@ -239,7 +239,7 @@ module FFOLCompleteness where
     app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
     app prf prf' = λ w γ → prf w γ w id-a (prf' w γ)
     -- Again, we don't write the _[_]p equalities as everything is in Prop
-vv                                                                      
+    
     -- ∀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)
diff --git a/FFOLInitial.agda b/FFOLInitial.agda
index 2bf0162..902ebd6 100644
--- a/FFOLInitial.agda
+++ b/FFOLInitial.agda
@@ -4,7 +4,7 @@ open import PropUtil
 
 module FFOLInitial where
 
-  open import FinitaryFirstOrderLogic
+  open import FFOL
   open import Agda.Primitive
   open import ListUtil
 
diff --git a/InfinitaryFirstOrderLogic.agda b/IFOL.agda
similarity index 97%
rename from InfinitaryFirstOrderLogic.agda
rename to IFOL.agda
index c32b5a9..fe3108e 100644
--- a/InfinitaryFirstOrderLogic.agda
+++ b/IFOL.agda
@@ -1,15 +1,14 @@
-{-# OPTIONS --prop #-}
+{-# OPTIONS --prop --rewriting #-}
 
-open import Agda.Builtin.Nat
+open import PropUtil
 
-module InfinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
+module IFOL (Term : Set) (R : Nat → Set) where
 
-  open import PropUtil
   open import ListUtil
 
   data Args : Nat → Set where
     zero : Args 0
-    next : {n : Nat} → Args n → Term → Args (suc n)
+    next : {n : Nat} → Args n → Term → Args (succ n)
     
   data Form : Set where
     Rel : {n : Nat} → R n → (Args n) → Form
diff --git a/InfinitaryFirstOrderKripke.agda b/IFOLKripke.agda
similarity index 93%
rename from InfinitaryFirstOrderKripke.agda
rename to IFOLKripke.agda
index ced5a87..a8ad576 100644
--- a/InfinitaryFirstOrderKripke.agda
+++ b/IFOLKripke.agda
@@ -1,12 +1,11 @@
-{-# OPTIONS --prop #-}
+{-# OPTIONS --prop --rewriting #-}
 
-open import Agda.Builtin.Nat
+open import PropUtil
 
-module InfinitaryFirstOrderKripke (Term : Set) (R : Nat → Set) where
+module IFOLKripke (Term : Set) (R : Nat → Set) where
 
   open import ListUtil
-  open import PropUtil
-  open import InfinitaryFirstOrderLogic Term R
+  open import IFOL Term R
 
   private
     variable
diff --git a/InfinitaryFirstOrderKripkeGeneral.agda b/IFOLNormalization.agda
similarity index 92%
rename from InfinitaryFirstOrderKripkeGeneral.agda
rename to IFOLNormalization.agda
index 18a5d59..42f6e2b 100644
--- a/InfinitaryFirstOrderKripkeGeneral.agda
+++ b/IFOLNormalization.agda
@@ -1,14 +1,13 @@
-{-# OPTIONS --prop #-}
+{-# OPTIONS --prop --rewriting #-}
 
-open import Agda.Builtin.Nat hiding (zero)
+open import PropUtil hiding (zero)
 
-module InfinitaryFirstOrderKripkeGeneral (Term : Set) (R : Nat → Set) where
+module IFOLNormalization (Term : Set) (R : Nat → Set) where
 
-  open import ListUtil
-  open import PropUtil
-  open import InfinitaryFirstOrderLogic Term R using (Form; Args; Rel; _⇒_; _∧∧_; ⊤⊤; ∀∀; Con)
+  open import ListUtil hiding (zero)
+  open import IFOL Term R using (Form; Args; Rel; _⇒_; _∧∧_; ⊤⊤; ∀∀; Con)
 
-  open import InfinitaryFirstOrderKripke Term R using (Kripke)
+  open import IFOLKripke Term R using (Kripke)
 
   record Preorder (T : Set₀) : Set₁ where
     constructor order
@@ -78,7 +77,7 @@ module InfinitaryFirstOrderKripkeGeneral (Term : Set) (R : Nat → Set) where
   module NormalizationTests where
 
     {- Now using our records -}
-    open import InfinitaryFirstOrderLogic Term R hiding (Form; _⇒_; Con)
+    open import IFOL Term R hiding (Form; _⇒_; Con)
 
 
     ClassicNN : NormalAndNeutral
diff --git a/Readme.agda b/Readme.agda
index 5a5f249..907b46b 100644
--- a/Readme.agda
+++ b/Readme.agda
@@ -1,31 +1,31 @@
-{-# OPTIONS --prop #-}
+{-# OPTIONS --prop --rewriting #-}
 
 module Readme where
 
 -- We will use String as propositional variables
-open import Agda.Builtin.String using (String)
-open import ListUtil
+postulate String : Set
+{-# BUILTIN STRING String #-}
+
 open import PropUtil
+open import ListUtil
 
 -- We can use the basic propositional logic
-open import PropositionalLogic String
+open import ZOL 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 (zero $ next∈ $ next∈ zero∈) (app (zero $ next∈ zero∈) (zero zero∈)))))
+zol-ex : [] ⊢ ((Var "Q") ⇒ (Var "R")) ⇒ ((Var "P") ⇒ (Var "Q")) ⇒ (Var "P") ⇒ (Var "R")
+zol-ex = 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
 -- environnement where the statement doesn't hold
-
 ρ₀ : Env
 ρ₀ "P" = ⊥
 ρ₀ "Q" = ⊤
 ρ₀ _ = ⊥
-
-cex-d : ([] ⊢ (((Var "P") ⇒ (Var "Q")) ⇒ (Var "P"))) → ⊥
-cex-d h = ⟦ h ⟧ᵈ {ρ₀} tt λ x → tt
+zol-cex : ([] ⊢ (((Var "P") ⇒ (Var "Q")) ⇒ (Var "P"))) → ⊥
+zol-cex h = ⟦ h ⟧ᵈ {ρ₀} tt λ x → tt
 
 -- But this is not enough to show the non-provability of every non-provable statement.
 -- Let's take as an example Pierce formula : ((P → Q) → P) → P
@@ -52,7 +52,7 @@ TND→P tnd {P} {Q} pqp = dis (tnd {P}) (λ p → p) (λ np → pqp (λ p → ca
 -- that the Pierce formula cannot be proven
 
 -- We import the general definition of Kripke models
-open import PropositionalKripke String
+open import ZOLKripke String
 
 -- We will now create a specific Kripke model in which Pierce formula doesn't hold
 
@@ -116,9 +116,9 @@ module PierceDisproof where
   PierceNotProvable h = Pierce⊥w₁ (⟦ h ⟧ {w₁} tt)
   
   
-module GeneralizationInPropositionalLogic where
+module GeneralizationInZOL where
 
-  -- With Kripke models, we can even prove completeness
+  -- With Kripke models, we can even prove completeness of ZOL
   -- Using the Universal Kripke Model
   
   -- With a slightly different universal model (using normal and neutral forms),
@@ -130,7 +130,7 @@ module GeneralizationInPropositionalLogic where
 
   -- As all those proofs are really similar, we created a NormalizationFrame structure
   -- that computes most of the proofs with only a few lemmas
-  open import PropositionalKripkeGeneral String
+  open import ZOLNormalization String
 
   -- We now have access to quote and unquote functions with this
   u1 = NormalizationFrame.u NormalizationTests.Frame⊢
@@ -146,13 +146,13 @@ module GeneralizationInPropositionalLogic where
   u6 = NormalizationFrame.u NormalizationTests.Frame⊆
   q6 = NormalizationFrame.q NormalizationTests.Frame⊆
 
-module GeneralizationInInfinitaryFirstOrderLogic where
+module GeneralizationInIFOL where
 
   -- We also did implement infinitary first order logic
   -- (i.e. ∀ is like an infinitary ∧)
   -- The proofs works the same with only little modifications
 
-  open import InfinitaryFirstOrderKripkeGeneral String (λ n → String)
+  open import IFOLNormalization String (λ n → String)
 
   u1 = NormalizationFrame.u NormalizationTests.Frame⊢
   q1 = NormalizationFrame.q NormalizationTests.Frame⊢
@@ -168,4 +168,22 @@ module GeneralizationInInfinitaryFirstOrderLogic where
   q6 = NormalizationFrame.q NormalizationTests.Frame⊆
 
 
+module NormalizationInFFOL where
+
+  -- We also did an implementation of the negative fragment
+  -- of finitary first order logic (∀ is defined with context extension)
+
+  -- The algebra has been written in this file
+  -- There is also the class of Tarski models, written as an example
+  open import FFOL
+
+  -- We have also written the syntax (initial model)
+  -- (a lot of transport hell, but i did it !)
+  open import FFOLInitial
+
+  -- And now, we can finally write the class of Family and Presheaf models
+  -- and we can make the proof of completeness of the latter.
+  open import FFOLCompleteness
+
+
 
diff --git a/PropositionalLogic.agda b/ZOL.agda
similarity index 99%
rename from PropositionalLogic.agda
rename to ZOL.agda
index 4972b8d..dc88867 100644
--- a/PropositionalLogic.agda
+++ b/ZOL.agda
@@ -1,6 +1,6 @@
-{-# OPTIONS --prop #-}
+{-# OPTIONS --prop --rewriting #-}
 
-module PropositionalLogic (PV : Set) where
+module ZOL (PV : Set) where
 
   open import PropUtil
   open import ListUtil
diff --git a/PropositionalKripke.agda b/ZOLKripke.agda
similarity index 95%
rename from PropositionalKripke.agda
rename to ZOLKripke.agda
index d251664..7ebfb13 100644
--- a/PropositionalKripke.agda
+++ b/ZOLKripke.agda
@@ -1,10 +1,10 @@
-{-# OPTIONS --prop #-}
+{-# OPTIONS --prop --rewriting #-}
 
-module PropositionalKripke (PV : Set) where
+module ZOLKripke (PV : Set) where
 
   open import ListUtil
   open import PropUtil
-  open import PropositionalLogic PV
+  open import ZOL PV
 
   private
     variable
diff --git a/PropositionalKripkeGeneral.agda b/ZOLNormalization.agda
similarity index 93%
rename from PropositionalKripkeGeneral.agda
rename to ZOLNormalization.agda
index 9cf4716..b6339c9 100644
--- a/PropositionalKripkeGeneral.agda
+++ b/ZOLNormalization.agda
@@ -1,12 +1,12 @@
-{-# OPTIONS --prop #-}
+{-# OPTIONS --prop --rewriting #-}
 
-module PropositionalKripkeGeneral (PV : Set) where
+module ZOLNormalization (PV : Set) where
 
-  open import ListUtil
-  open import PropUtil
-  open import PropositionalLogic PV using (Form; Var; _⇒_; _∧∧_; ⊤⊤; Con)
+  open import ListUtil hiding (zero)
+  open import PropUtil hiding (zero)
+  open import ZOL PV using (Form; Var; _⇒_; _∧∧_; ⊤⊤; Con)
 
-  open import PropositionalKripke PV using (Kripke)
+  open import ZOLKripke PV using (Kripke)
 
   record Preorder (T : Set₀) : Set₁ where
     constructor order
@@ -75,7 +75,7 @@ module PropositionalKripkeGeneral (PV : Set) where
   module NormalizationTests where
 
     {- Now using our records -}
-    open import PropositionalLogic PV hiding (Form; Var; _⇒_; Con)
+    open import ZOL PV hiding (Form; Var; _⇒_; Con)
 
 
     ClassicNN : NormalAndNeutral
diff --git a/fol-subst.agda b/fol-subst.agda
deleted file mode 100644
index 6c0c2e1..0000000
--- a/fol-subst.agda
+++ /dev/null
@@ -1,138 +0,0 @@
-open import Data.Nat
-open import Relation.Binary.PropositionalEquality
-
-variable m n l : ℕ
-
-data Term : ℕ → Set where
-  zero : Term (suc n)
-  suc : Term n → Term (suc n)
-
-variable t u : Term n
-
-data Subst : ℕ → ℕ → Set where
-  ε : Subst n 0
-  _,_ : Subst m n → Term m → Subst m (suc n)
-
-suc-subst : Subst m n → Subst (suc m) n
-suc-subst ε = ε
-suc-subst (ts , t) = suc-subst ts , suc t
-
-id : Subst m m
-id {zero} = ε
-id {suc m} = (suc-subst (id {m})) , zero
-
-subst-t : Term m → Subst n m → Term n
-subst-t zero (us , u) = u
-subst-t (suc t) (us , u) = subst-t t us
-
-wk-t1 : Term n → Term (suc n)
-wk-t1 t = subst-t t (suc-subst id)
-
-subst-t1 : Term (suc n) → Term n → Term n
-subst-t1 t u = subst-t t (id , u)
-
-comp : Subst m l → Subst n m → Subst n l
-comp ε us = ε
-comp (ts , u) us = (comp ts us) , (subst-t u us)
-
-{-
-t [ suc-subst vs ] ≡ suc (t [vs ])
-suc-subst ts ∘ (us , t) ≡ ts ∘ us
-
-t [ id ] ≡ t
-t [ us ∘ vs ] ≡ t [ us ] [ vs ]
-
-ts ∘ id ≡ ts
-id ∘ ts ≡ ts
-(ts ∘ us) ∘ vs ≡ ts ∘ (us ∘ vs)
-
--}
-
-
-{-
-wk-t : (l : ℕ) → Term (l + n) → Term (suc (l + n))
-wk-t zero t = suc t
-wk-t (suc l) zero = zero
-wk-t (suc l) (suc t) = suc (wk-t l t)
-
-subst-t : (l : ℕ) → Term (suc (l + n)) → Term n → Term (l + n)
-subst-t zero zero u = u
-subst-t zero (suc t) u = t
-subst-t (suc l) zero u = zero
-subst-t (suc l) (suc t) u = suc (subst-t l t u)
-
-infix 15 _⇒_
-
-data Form : ℕ → Set where
-  _⇒_ : Form n → Form n → Form n
-  ∀F : Form (suc n) → Form n
-  P : Term n → Form n
---  R : Term n → Term n → Form n
-
-wk-F : (l : ℕ) → Form (l + n) → Form (suc (l + n))
-wk-F l (A ⇒ B) = wk-F l A ⇒ wk-F l B
-wk-F l (∀F A) = ∀F (wk-F (suc l) A)
-wk-F l (P t) = P (wk-t l t)
-
-subst-F : (l : ℕ) → Form (suc (l + n)) → Term n → Form (l + n)
-subst-F l (A ⇒ B) u = subst-F l A u ⇒ subst-F l B u
-subst-F l (∀F A) u = ∀F (subst-F (suc l) A u)
-subst-F l (P t) u = P (subst-t l t u)
-
-infix 10 _▷_
-
-data Con : ℕ → Set where
-  • : Con n
-  _▷_ : Con n → Form n → Con n
-
-wk-C : (l : ℕ) → Con (l + n) → Con (suc (l + n))
-wk-C l • = •
-wk-C l (Γ ▷ A) = wk-C l Γ ▷ wk-F l A
-
-variable Γ Δ : Con n
-variable A B C : Form n
-
-infix 5 _⊢_ 
-
-data _⊢_ : Con n → Form n → Set where
-  zero : Γ ▷ A ⊢ A
-  suc : Γ ⊢ A → Γ ▷ B ⊢ A
-  lam : Γ ▷ A ⊢ B → Γ ⊢ A ⇒ B
-  app : Γ ⊢ A ⇒ B → Γ ⊢ A → Γ ⊢ B
-  Lam : (wk-C zero Γ) ⊢ A → Γ ⊢ ∀F A
-  App : Γ ⊢ ∀F A → (t : Term _) → Γ ⊢ subst-F zero A t
-
-{-
--- (A ⇒ ∀ x . P x) ⇒ ∀ x . A → P x
-
--- A ≡ A [ wk ][ < t > ] 
-
-wk-subst : subst l (wk l A) t ≡ t
-
-wk-subst :  (A [ wk ]F) s[ < t > ]F ≡ A
-wk-subst = {!!}
-
-example : • ⊢ (A ⇒ (∀F (P zero))) ⇒ (∀F (A [ wk ]F) ⇒ P zero)
-example {A = A} = lam (lam (App (app (suc zero) 
-         (subst (λ X → (• ▷ A ⇒ ∀F (P zero)) ▷ ∀F (A [ wk ]F) ⊢ X)
-           (wk-subst {A = A}) (App zero zero))) zero))
-
-
--- (∀ x ∀ y . A(x,y)) ⇒ ∀ y ∀ x . A(y,x)
--- (A ⇒ ∀ x . B(x)) ⇒ ∀ x . A ⇒ B(x)   
--- ∀ x y . A(x,y) ⇒ ∀ x . A(x,x)
--- ∀ x . A (x) ⇒ ∀ x y . A(x)
--- (((∀ x . A (x)) ⇒ B)⇒ B) ⇒ ∀ x . ((A (x) ⇒ B) ⇒ B)
-
--}
-
---eqq : (suc (suc (l + n))) ≡ suc l + suc n
-eqq : (suc (l + n)) ≡ l + suc n
-eqq = {!!}
-
-eq : ∀ {t : Term (suc (l + n))}{u : Term n} →
---   subst-t {n = suc n} l (subst Term (eqq {l = l}{n = n})(wk-t {n = n} (suc l) t)) (wk-t zero u)
-   subst-t {n = suc n} l (subst Term (eqq {l = (suc l)}{n = n})(wk-t {n = n} (suc l) t)) (wk-t zero u)
-   ≡ {! subst-t l t   !} -- subst-t l (wk-t (suc l) t) u ≡ subst-t l t u
-eq = {!!}
--}
diff --git a/fol.agda b/fol.agda
deleted file mode 100644
index 89f8f49..0000000
--- a/fol.agda
+++ /dev/null
@@ -1,124 +0,0 @@
-open import Data.Nat
-open import Relation.Binary.PropositionalEquality
-
-variable m n l : ℕ
-
-_$_ : {A B : Set} → (A → B) → A → B
-f $ x = f x
-infixr 1 _$_
-
-data Term : ℕ → Set where
-  zero : Term (suc n)
-  suc : Term n → Term (suc n)
-
-variable t u : Term n
-
-wk-t : (l : ℕ) → Term (l + n) → Term (suc (l + n))
-wk-t zero t = suc t
-wk-t (suc l) zero = zero
-wk-t (suc l) (suc t) = suc (wk-t l t)
-
-subst-t : (l : ℕ) → Term (suc (l + n)) → Term n → Term (l + n)
-subst-t zero zero u = u
-subst-t zero (suc t) u = t
-subst-t (suc l) zero u = zero
-subst-t (suc l) (suc t) u = suc (subst-t l t u)
-
-infix 15 _⇒_
-
-data Form : ℕ → Set where
-  _⇒_ : Form n → Form n → Form n
-  ∀F : Form (suc n) → Form n
-  P : Term n → Form n
---  R : Term n → Term n → Form n
-
-wk-F : (l : ℕ) → Form (l + n) → Form (suc (l + n))
-wk-F l (A ⇒ B) = wk-F l A ⇒ wk-F l B
-wk-F l (∀F A) = ∀F (wk-F (suc l) A)
-wk-F l (P t) = P (wk-t l t)
-
-subst-F : (l : ℕ) → Form (suc (l + n)) → Term n → Form (l + n)
-subst-F l (A ⇒ B) u = subst-F l A u ⇒ subst-F l B u
-subst-F l (∀F A) u = ∀F (subst-F (suc l) A u)
-subst-F l (P t) u = P (subst-t l t u)
-
-infix 10 _▷_
-
-data Con : ℕ → Set where
-  • : Con n
-  _▷_ : Con n → Form n → Con n
-
-wk-C : (l : ℕ) → Con (l + n) → Con (suc (l + n))
-wk-C l • = •
-wk-C l (Γ ▷ A) = wk-C l Γ ▷ wk-F l A
-
-variable Γ Δ : Con n
-variable A B C : Form n
-
-infix 5 _⊢_ 
-
-data _⊢_ : Con n → Form n → Set where
-  zero : Γ ▷ A ⊢ A
-  suc : Γ ⊢ A → Γ ▷ B ⊢ A
-  lam : Γ ▷ A ⊢ B → Γ ⊢ A ⇒ B
-  app : Γ ⊢ A ⇒ B → Γ ⊢ A → Γ ⊢ B
-  Lam : (wk-C zero Γ) ⊢ A → Γ ⊢ ∀F A
-  App : Γ ⊢ ∀F A → (t : Term _) → Γ ⊢ subst-F zero A t
-
-
-
--- A ≡ A [ wk ][ < t > ] 
-
-wk-substt : {t : Term (l + n)} → subst-t l (wk-t l t) u ≡ t
-wk-substt {zero} = refl
-wk-substt {suc l} {t = zero} = refl
-wk-substt {suc l} {n} {t = suc t} = cong (λ t → suc t) (wk-substt {l})
-
-wk-substf : {A : Form (l + n)} → subst-F l (wk-F l A) u ≡ A
-wk-substf {A = A ⇒ A₁} = cong₂ (λ B B₁ → B ⇒ B₁) wk-substf wk-substf
-wk-substf {A = ∀F A} = cong (λ B → ∀F B) wk-substf
-wk-substf {l = l} {A = P x} = cong (λ t → P t) (wk-substt {l})
-
--- (A ⇒ ∀ x . P x) ⇒ ∀ x . A → P x
-example : • ⊢ (A ⇒ (∀F (P zero))) ⇒ (∀F (wk-F 0 A) ⇒ P zero)
-example {A = A} = lam (lam (App (app (suc zero) (subst (λ Φ → (• ▷ A ⇒ ∀F (P zero)) ▷ ∀F (wk-F 0 A) ⊢ Φ) wk-substf (App zero zero))) zero))
-
--- (∀ x ∀ y . A(x,y)) ⇒ ∀ y ∀ x . A(y,x)
-ex1 : {A : Form 2} → • ⊢ (∀F (∀F A)) ⇒ (∀F (∀F A))
-ex1 = lam zero
--- (A ⇒ ∀ x . B(x)) ⇒ ∀ x . A ⇒ B(x)
--- y → ∀x B(x) ==> y → ∀ x B(y)
-eq' : {l n : ℕ} → (l + suc n) ≡ (suc (l + n))
-eq : {l n : ℕ} → (1 + (l + (suc n))) ≡ (suc (suc (l + n)))
---eq {zero} {zero} = refl
---eq {zero} {suc n} = cong suc (eq {l = 0})
---eq {suc l} = cong suc (eq {l = l})
-lm-t : {l n : ℕ} → {t : Term (l + suc n)} → subst-t {n = suc n} l (subst Term (sym eq) (wk-t (suc l) (subst Term eq' t)) ) zero ≡ t
-lm-F : {l n : ℕ} → {A : Form (l + suc n)} → subst-F {n = suc n} l (subst Form (sym eq) (wk-F (suc l) (subst Form eq' A)) ) zero ≡ A
-lm-t {t = t} = {!!}
-lm-F = {!!}
-{-
-lm-F : {A : Form 1} → subst-F 0 (wk-F 1 A) zero ≡ A
-lm-F {A ⇒ A₁} = cong₂ (λ B B₁ → B ⇒ B₁) lm-F lm-F
-lm-F {∀F A} = cong (λ B → ∀F B) {!lm-F!}
-lm-F {P x} = cong (λ t → P t) lm-t
--}
-ex2 : {A : Form 0} → {B : Form 1} → • ⊢ ((A ⇒ (∀F B)))⇒(∀F ((wk-F 0 A) ⇒ B))
-ex2 {A = A} {B = B} = lam $ Lam $ lam $ subst (λ C → (• ▷ wk-F zero A ⇒ ∀F (wk-F 1 B)) ▷ wk-F 0 A ⊢ C) lm-F (((App (app (suc zero) zero) zero)))
--- lam (Lam (lam ( subst (λ Φ → (• ▷ wk-F zero A ⇒ ∀F (wk-F 1 B)) ▷ wk-F 0 A ⊢ Φ) (wk-substf {l = 0}) (App (app (suc (suc lm)) (app (suc zero) zero)) zero))))
--- ∀ x y . A(x,y) ⇒ ∀ x . A(x,x)
--- ∀ x . A (x) ⇒ ∀ x y . A(x)
--- (((∀ x . A (x)) ⇒ B)⇒ B) ⇒ ∀ x . ((A (x) ⇒ B) ⇒ B)
-
-{-
---eqq : (suc (suc (l + n))) ≡ suc l + suc n
-eqq : (suc (l + n)) ≡ l + suc n
-eqq = {!!}
-
-eq : ∀ {t : Term (suc (l + n))}{u : Term n} →
---   subst-t {n = suc n} l (subst Term (eqq {l = l}{n = n})(wk-t {n = n} (suc l) t)) (wk-t zero u)
-   subst-t {n = suc n} l (subst Term (eqq {l = (suc l)}{n = n})(wk-t {n = n} (suc l) t)) (wk-t zero u)
-   ≡ {! subst-t l t   !} -- subst-t l (wk-t (suc l) t) u ≡ subst-t l t u
-eq = {!!}
-
--}