From 2b99239d3d7c008e13fd2b22cafdb52e2ef88d55 Mon Sep 17 00:00:00 2001
From: Max New <maxsnew@gmail.com>
Date: Thu, 18 May 2023 12:30:47 -0400
Subject: [PATCH] Begin porting syntax to GTT and new Context rep

---
 .../guarded-cubical/Syntax/Types.agda         | 128 ++++++++----------
 1 file changed, 53 insertions(+), 75 deletions(-)

diff --git a/formalizations/guarded-cubical/Syntax/Types.agda b/formalizations/guarded-cubical/Syntax/Types.agda
index d237c2f..610c547 100644
--- a/formalizations/guarded-cubical/Syntax/Types.agda
+++ b/formalizations/guarded-cubical/Syntax/Types.agda
@@ -9,8 +9,6 @@ open import Common.Later hiding (next)
 
 module Syntax.Types  where
 
-
-
 open import Cubical.Foundations.Prelude renaming (comp to compose)
 open import Cubical.Data.Nat hiding (_·_) renaming (ℕ to Nat)
 open import Cubical.Relation.Nullary
@@ -20,12 +18,9 @@ open import Cubical.Foundations.Isomorphism
 open import Cubical.Data.List
   using (List ; length ; map ; _++_ ; cons-inj₁ ; cons-inj₂)
   renaming ([] to · ; _∷_ to _::_)
-
 open import Cubical.Data.Empty renaming (rec to exFalso)
 
-
-import Syntax.DeBruijnCommon
-
+open import Syntax.Context as Context
 
 private
  variable
@@ -37,33 +32,27 @@ private
 data Interval : Type where
   l r : Interval
 
-data IntExt : Type where
-  Int Ext : IntExt
-
 data iCtx : Type where
   Empty : iCtx
   Full : iCtx
 
 
-private
-  variable
-    α : IntExt
+data Ty : iCtx -> Type
 
-data Ty : {α : IntExt} -> iCtx -> Type
+ty-endpt : Interval -> Ty Full -> Ty Empty
 
-ty-endpt : ∀ {α} -> Interval -> Ty {α} Full -> Ty {α} Empty
+_⊑_ : Ty Empty -> Ty Empty -> Type
+A ⊑ B = Σ[ c ∈ Ty Full ] ((ty-endpt l c ≡ A) × (ty-endpt r c ≡ B))
 
 data Ty where
-  nat : ∀ {α Ξ} -> Ty {α} Ξ
-  dyn : ∀ {α Ξ} -> Ty {α} Ξ
-  _⇀_ : ∀ {α Ξ} -> Ty {α} Ξ -> Ty {α} Ξ -> Ty {α} Ξ
-  inj-nat : ∀ {α} -> Ty {α} Full
-  inj-arr : ∀ {α} -> Ty {α} Full
-  comp : ∀ {α} -> (c : Ty {α} Full) -> (d : Ty {α} Full) ->
-    (ty-endpt l c ≡ ty-endpt r d) -> Ty {α} Full
-  -- order-set : isSet (Ty Full)
-  ▹ : ∀ {Ξ} -> Ty {Int} Ξ -> Ty {Int} Ξ
-
+  nat : ∀ {Ξ} -> Ty Ξ
+  dyn : ∀ {Ξ} -> Ty Ξ
+  _⇀_ : ∀ {Ξ} -> Ty Ξ -> Ty Ξ -> Ty Ξ
+  inj-nat : Ty Full
+  inj-arr : Ty Full
+  comp : ∀ (c : Ty Full) -> (d : Ty Full) ->
+    (ty-endpt l c ≡ ty-endpt r d) -> Ty Full
+  -- isProp⊑ : ∀ {A B : Ty }
 
 ty-endpt p nat = nat
 ty-endpt p dyn = dyn
@@ -74,26 +63,27 @@ ty-endpt l inj-arr = (dyn ⇀ dyn) -- inj-arr : (dyn -> dyn) ⊑ dyn
 ty-endpt r inj-arr = dyn
 ty-endpt l (comp c d _) = ty-endpt l d
 ty-endpt r (comp c d _) = ty-endpt r c
-ty-endpt p (â–¹ A) = â–¹ (ty-endpt p A)
 
 
-_[_/i] : ∀ {α} -> Ty {α} Full -> Interval -> Ty {α} Empty
+
+
+_[_/i] : Ty Full -> Interval -> Ty Empty
 c [ p /i] = ty-endpt p c
 
-ty-left : ∀ {α} -> Ty {α} Full -> Ty Empty
+ty-left : Ty Full -> Ty Empty
 ty-left = ty-endpt l
 
-ty-right : ∀ {α} -> Ty {α} Full -> Ty Empty
+ty-right : Ty Full -> Ty Empty
 ty-right = ty-endpt r
 
-CompTyRel : ∀ {α} -> Type
-CompTyRel {α} = Σ (Ty {α} Full × Ty Full)
+CompTyRel : Type
+CompTyRel = Σ (Ty Full × Ty Full)
   λ { (c' , c) -> (ty-left c') ≡ (ty-right c) }
 
-CompTyRel-comp : ∀ {α} -> CompTyRel {α} -> Ty Full
+CompTyRel-comp : CompTyRel -> Ty Full
 CompTyRel-comp ((c' , c) , pf) = comp c' c pf
 
-CompTyRel-endpt : ∀ {α} -> Interval -> CompTyRel {α} -> Ty Full
+CompTyRel-endpt : Interval -> CompTyRel -> Ty Full
 CompTyRel-endpt l ((c , d) , _) = c
 CompTyRel-endpt r ((c , d) , _) = d
 
@@ -101,69 +91,57 @@ CompTyRel-endpt r ((c , d) , _) = d
 
 
 -- Given a "normal" type A, view it as its reflexivity precision derivation c : A ⊑ A.
-ty-refl : Ty {α} Empty -> Ty {α} Full
+ty-refl : Ty Empty -> Ty Full
 ty-refl nat = nat
 ty-refl dyn = dyn
 ty-refl (Ai ⇀ Ao) = ty-refl Ai ⇀ ty-refl Ao
-ty-refl (â–¹ A) = â–¹ (ty-refl A)
-
-ty-endpt-refl : {A : Ty {α} Empty} -> (p : Interval) -> ty-endpt p (ty-refl A) ≡ A
-ty-endpt-refl {_} {nat} p = refl
-ty-endpt-refl {_} {dyn} p = refl
-ty-endpt-refl {_} {A ⇀ B} p = cong₂ _⇀_ (ty-endpt-refl p) (ty-endpt-refl p)
-ty-endpt-refl {_} {â–¹ A} p = cong â–¹ (ty-endpt-refl p)
-
-
-_⊑_ : Ty {α} Empty -> Ty Empty -> Type
-A ⊑ B = Σ[ c ∈ Ty Full ] ((ty-left c ≡ A) × (ty-right c ≡ B))
-
 
+ty-endpt-refl : {A : Ty Empty} -> (p : Interval) -> ty-endpt p (ty-refl A) ≡ A
+ty-endpt-refl {nat} p = refl
+ty-endpt-refl {dyn} p = refl
+ty-endpt-refl {A ⇀ B} p = cong₂ _⇀_ (ty-endpt-refl p) (ty-endpt-refl p)
 
 -- ############### Contexts ###############
+open Ctx
 
-
-Ctx : ∀ {α : IntExt} -> iCtx -> Type
-Ctx {α} Ξ = List (Ty {α} Ξ)
-
-
+TyCtx : iCtx → Type (ℓ-suc ℓ-zero)
+TyCtx Ξ = Ctx (Ty Ξ)
 
 -- Endpoints of a full context
-ctx-endpt : (p : Interval) -> Ctx {α} Full -> Ctx Empty
-ctx-endpt p = map (ty-endpt p)
-
-CompCtx : (Δ Γ : Ctx {α} Full) -> (pf : ctx-endpt l Δ ≡ ctx-endpt r Γ) ->
-  Ctx {α} Full
-CompCtx Δ Γ pf = {!!}
-
-
-
+ctx-endpt : (p : Interval) -> TyCtx Full -> TyCtx Empty
+ctx-endpt p = Context.map (ty-endpt p)
 
+CompCtx : (Δ Γ : TyCtx Full)
+        -> (pf : ctx-endpt l Δ ≡ ctx-endpt r Γ)
+        -> TyCtx Full
+CompCtx Δ Γ pf .var = Δ .var
+CompCtx Δ Γ pf .isFinSetVar = Δ .isFinSetVar
+CompCtx Δ Γ pf .el x = comp (Δ .el x)
+                            (Γ .el (transport (cong var pf) x))
+                            λ i → pf i .el (transport-filler (cong var pf) x i)
 
--- "Contains" relation stating that a context Γ contains a type T
-data _∋_ : ∀ {Ξ} -> Ctx {α} Ξ -> Ty {α} Ξ -> Set where
-  vz : ∀ {Ξ Γ S} -> _∋_ {α} {Ξ} (S :: Γ) S
-  vs : ∀ {Ξ Γ S T} (x : _∋_ {α} {Ξ} Γ T) -> (S :: Γ ∋ T)
+-- -- "Contains" relation stating that a context Γ contains a type T
+-- data _∋_ : ∀ {Ξ} -> Ctx Ξ -> Ty Ξ -> Set where
+--   vz : ∀ {Ξ Γ S} -> _∋_ {Ξ} (S :: Γ) S
+--   vs : ∀ {Ξ Γ S T} (x : _∋_ {Ξ} Γ T) -> (S :: Γ ∋ T)
 
-infix 4 _∋_
+-- infix 4 _∋_
 
-∋-ctx-endpt : {Γ : Ctx {α} Full} {c : Ty Full} -> (p : Interval) ->
-  (Γ ∋ c) -> ((ctx-endpt p Γ) ∋ (ty-endpt p c))
-∋-ctx-endpt p vz = vz
-∋-ctx-endpt p (vs Γ∋c) = vs (∋-ctx-endpt p Γ∋c)
+-- ∋-ctx-endpt : {Γ : Ctx Full} {c : Ty Full} -> (p : Interval) ->
+--   (Γ ∋ c) -> ((ctx-endpt p Γ) ∋ (ty-endpt p c))
+-- ∋-ctx-endpt p vz = vz
+-- ∋-ctx-endpt p (vs Γ∋c) = vs (∋-ctx-endpt p Γ∋c)
 
 
 
 -- View a "normal" typing context Γ as a type precision context where the derivation
 -- corresponding to each type A in Γ is just the reflexivity precision derivation A ⊑ A.
-ctx-refl : Ctx {α} Empty -> Ctx Full
-ctx-refl = map ty-refl
---ctx-refl · = ·
---ctx-refl (A :: Γ) = ty-refl A :: ctx-refl Γ
+ctx-refl : TyCtx Empty -> TyCtx Full
+ctx-refl = Context.map ty-refl
 
 -- For a given typing context, the endpoints of the corresponding reflexivity precision
 -- context are the typing context itself.
-ctx-endpt-refl : {Γ : Ctx {α} Empty} -> (p : Interval) -> ctx-endpt p (ctx-refl Γ) ≡ Γ
-ctx-endpt-refl {_} {·} p = refl
-ctx-endpt-refl {_} {A :: Γ} p = cong₂ _::_  (ty-endpt-refl p) (ctx-endpt-refl p)
+ctx-endpt-refl : {Γ : TyCtx Empty} -> (p : Interval) -> ctx-endpt p (ctx-refl Γ) ≡ Γ
+ctx-endpt-refl {Γ} p = Ctx≡ _ _ refl (funExt λ x → ty-endpt-refl {A = Γ .el x} p)
 
 
-- 
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