Dyn as a Poset

formalizations/guarded-cubical/Semantics/Concrete/Dyn.agda

+{-# OPTIONS --rewriting --guarded #-}
+
+ -- to allow opening this module in other files while there are still holes
+{-# OPTIONS --allow-unsolved-metas #-}
+
+
+open import Common.Later
+
+module Semantics.Concrete.Dyn (k : Clock) where
+
+open import Cubical.Foundations.Prelude
+open import Cubical.Foundations.HLevels
+open import Cubical.Foundations.Isomorphism
+open import Cubical.Foundations.Structure
+open import Cubical.Foundations.Equiv
+open import Cubical.Foundations.Univalence
+
+open import Cubical.Relation.Binary
+open import Cubical.Data.Nat renaming (ℕ to Nat)
+open import Cubical.Data.Sum
+open import Cubical.Data.Unit
+open import Cubical.Data.Empty
+
+open import Common.LaterProperties
+--open import Common.Preorder.Base
+--open import Common.Preorder.Monotone
+--open import Common.Preorder.Constructions
+open import Semantics.Lift k
+-- open import Semantics.Concrete.LiftPreorder k
+
+open import Cubical.Relation.Binary.Poset
+open import Common.Poset.Convenience
+open import Common.Poset.Constructions
+open import Common.Poset.Monotone
+
+
+open import Semantics.LockStepErrorOrdering k
+
+open BinaryRelation
+open LiftPoset
+
+private
+  variable
+    ℓ ℓ' : Level
+
+  ▹_ : Type ℓ → Type ℓ
+  ▹_ A = ▹_,_ k A
+
+
+data Dyn' (D : ▹ Poset ℓ ℓ') : Type (ℓ-max ℓ ℓ') where
+  nat : Nat -> Dyn' D
+  fun : ▸ (λ t → MonFun (D t) (𝕃 (D t))) -> Dyn' D
+
+Dyn'-iso : (D : ▹ Poset ℓ ℓ') -> Iso (Dyn' D) (Nat ⊎ (▸ (λ t → MonFun (D t) (𝕃 (D t)))))
+Dyn'-iso D = iso
+  (λ { (nat n) → inl n ; (fun f~) → inr f~})
+  (λ { (inl n) → nat n ; (inr f~) → fun f~})
+  (λ { (inl n) → refl ;  (inr f~) → refl})
+  (λ { (nat x) → refl ;  (fun x) → refl})
+
+
+DynP' :
+    (D : ▹ Poset ℓ-zero ℓ-zero) -> Poset ℓ-zero ℓ-zero
+DynP' D = Dyn' D ,
+    posetstr order
+      (isposet isSetDynP' dyn-ord-prop dyn-ord-refl dyn-ord-trans dyn-ord-antisym)
+    where
+      order : Dyn' D → Dyn' D → Type
+      order (nat n) (nat m) = (n ≡ m)
+      order (fun f~) (fun g~) = ▸ λ t → (f~ t) ≤mon (g~ t)
+      order _ _ = ⊥
+
+      isSetDynP' : isSet (Dyn' D)
+      isSetDynP' = isSetRetract
+        (Iso.fun (Dyn'-iso D)) (Iso.inv (Dyn'-iso D)) (Iso.leftInv (Dyn'-iso D))
+        (isSet⊎ isSetℕ (isSet▸ λ t -> MonFunIsSet))
+
+      dyn-ord-refl : isRefl order
+      dyn-ord-refl (nat n) = refl
+      dyn-ord-refl (fun f~) = λ t → ≤mon-refl (f~ t)
+
+      dyn-ord-prop : isPropValued order
+      dyn-ord-prop (nat n) (nat m) = isSetℕ n m
+      dyn-ord-prop (fun f~) (fun g~) = isProp▸ (λ t -> ≤mon-prop (f~ t) (g~ t))
+
+      dyn-ord-trans : isTrans order
+      dyn-ord-trans (nat n1) (nat n2) (nat n3) n1≡n2 n2≡n3 =
+        n1≡n2 ∙ n2≡n3
+      dyn-ord-trans (fun f1~) (fun f2~) (fun f3~) H1 H2 =
+        λ t → ≤mon-trans (f1~ t) (f2~ t) (f3~ t) (H1 t) (H2 t)
+
+      dyn-ord-antisym : isAntisym order
+      dyn-ord-antisym (nat n) (nat m) n≡m m≡n = cong nat n≡m
+      dyn-ord-antisym (fun f~) (fun g~) d≤d' d'≤d =
+        cong fun (eq▸ f~ g~ λ t -> ≤mon-antisym (f~ t) (g~ t) (d≤d' t) (d'≤d t))
+
+
+DynP : Poset ℓ-zero ℓ-zero
+DynP = fix DynP'
+
+unfold-DynP : DynP ≡ DynP' (next DynP)
+unfold-DynP = fix-eq DynP'
+
+unfold-⟨DynP⟩ : ⟨ DynP ⟩ ≡ ⟨ DynP' (next DynP) ⟩
+unfold-⟨DynP⟩ = λ i → ⟨ unfold-DynP i ⟩
+
+unfold-DynP-rel : PathP (λ i -> {!lift (unfold-⟨DynP⟩ i)!}) (rel DynP) (rel (DynP' (next DynP)))
+unfold-DynP-rel = {!!}
+
+
+-- Converting from the underlying set of DynP' to the underlying
+-- set of DynP
+DynP'→DynP : ⟨ DynP' (next DynP) ⟩ -> ⟨ DynP ⟩
+DynP'→DynP d = transport (sym (λ i -> ⟨ unfold-DynP i ⟩)) d
+
+DynP→DynP' : ⟨ DynP ⟩ -> ⟨ DynP' (next DynP) ⟩
+DynP→DynP' d = transport (λ i → ⟨ unfold-DynP i ⟩) d
+
+
+rel-DynP'→rel-DynP : ∀ d1 d2 ->
+  rel (DynP' (next DynP)) d1 d2 ->
+  rel DynP (DynP'→DynP d1) (DynP'→DynP d2)
+rel-DynP'→rel-DynP d1 d2 d1≤d2 = transport
+  (λ i → rel (unfold-DynP (~ i))
+    (transport-filler (λ j → ⟨ unfold-DynP (~ j) ⟩) d1 i)
+    (transport-filler (λ j → ⟨ unfold-DynP (~ j) ⟩) d2 i))
+  d1≤d2
+
+rel-DynP→rel-DynP' : ∀ d1 d2 ->
+  rel DynP d1 d2 ->
+  rel (DynP' (next DynP)) (DynP→DynP' d1) (DynP→DynP' d2)
+rel-DynP→rel-DynP' d1 d2 d1≤d2 = transport
+  (λ i → rel (unfold-DynP i)
+    (transport-filler (λ j -> ⟨ unfold-DynP j ⟩) d1 i)
+    (transport-filler (λ j -> ⟨ unfold-DynP j ⟩) d2 i))
+  d1≤d2
+
+DynP-equiv : PosetEquiv DynP (DynP' (next DynP))
+DynP-equiv = pathToEquiv unfold-⟨DynP⟩ ,
+             makeIsPosetEquiv (pathToEquiv unfold-⟨DynP⟩)
+               (λ d1 d2 d1≤d2 -> rel-DynP→rel-DynP' d1 d2 d1≤d2)
+               (λ d1 d2 d1≤d2 -> {!rel-DynP'→rel-DynP d1 d2 d1≤d2!})
+
+
+
+
+
+InjNat : ⟨ ℕ ==> DynP ⟩
+InjNat = record {
+  f = λ n -> DynP'→DynP (nat n) ;
+  isMon = λ {n} {m} n≡m ->
+    rel-DynP'→rel-DynP (nat n) (nat m) n≡m }
+
+
+InjArr : ⟨ (DynP ==> 𝕃 DynP) ==> DynP ⟩
+InjArr = record {
+  f = λ f -> DynP'→DynP (fun (next f)) ;
+  isMon = λ {f1} {f2} f1≤f2 ->
+    rel-DynP'→rel-DynP (fun (next f1)) (fun (next f2)) λ t -> f1≤f2 }
+
+
+
+
+
+
+
+
+{-
+-- Poset Structure on Dyn
+module DynPoset (ℓ ℓ' : Level) where
+
+  DynP' :
+    (D : ▹ Poset ℓ ℓ') -> Poset (ℓ-max ℓ ℓ') (ℓ-max ℓ ℓ')
+  DynP' D = Dyn' D ,
+    posetstr order
+      (isposet isSetDynP' dyn-ord-prop dyn-ord-refl dyn-ord-trans dyn-ord-antisym)
+    where
+      order : Dyn' D → Dyn' D → Type (ℓ-max ℓ ℓ')
+      order (nat n) (nat m) = Lift (n ≡ m)
+      order (fun f~) (fun g~) = ▸ λ t → (f~ t) ≤mon (g~ t)
+      order _ _ = ⊥*
+
+      isSetDynP' : isSet (Dyn' D)
+      isSetDynP' = isSetRetract
+        (Iso.fun (Dyn'-iso D)) (Iso.inv (Dyn'-iso D)) (Iso.leftInv (Dyn'-iso D))
+        (isSet⊎ isSetℕ (isSet▸ λ t -> MonFunIsSet))
+
+      dyn-ord-refl : isRefl order
+      dyn-ord-refl (nat n) = lift refl
+      dyn-ord-refl (fun f~) = λ t → ≤mon-refl (f~ t)
+
+      dyn-ord-prop : isPropValued order
+      dyn-ord-prop (nat n) (nat m) = isOfHLevelLift 1 (isSetℕ n m)
+      dyn-ord-prop (fun f~) (fun g~) = isProp▸ (λ t -> ≤mon-prop (f~ t) (g~ t))
+
+      dyn-ord-trans : isTrans order
+      dyn-ord-trans (nat n1) (nat n2) (nat n3) (lift n1≡n2) (lift n2≡n3) =
+        lift (n1≡n2 ∙ n2≡n3)
+      dyn-ord-trans (fun f1~) (fun f2~) (fun f3~) H1 H2 =
+        λ t → ≤mon-trans (f1~ t) (f2~ t) (f3~ t) (H1 t) (H2 t)
+
+      dyn-ord-antisym : isAntisym order
+      dyn-ord-antisym (nat n) (nat m) (lift n≡m) (lift m≡n) = cong nat n≡m
+      dyn-ord-antisym (fun f~) (fun g~) d≤d' d'≤d =
+        cong fun (eq▸ f~ g~ λ t -> ≤mon-antisym (f~ t) (g~ t) (d≤d' t) (d'≤d t))
+
+
+  DynP : Poset ? ?
+  DynP = fix DynP'
+
+
+  
+  unfold-DynP : DynP ≡ DynP' (next DynP)
+  unfold-DynP = fix-eq DynP'
+
+
+  unfold-⟨DynP⟩ : {ℓ ℓ' : Level} -> ⟨ DynP ⟩ ≡ ⟨ DynP' (next (DynP {ℓ} {ℓ'})) ⟩
+  unfold-⟨DynP⟩ {ℓ} {ℓ'} = λ i → ⟨ unfold-DynP {ℓ} {ℓ'} i ⟩
+
+
+  -- Converting from the underlying set of DynP' to the underlying
+  -- set of DynP
+  DynP'→DynP : ⟨ DynP' (next DynP) ⟩ -> ⟨ DynP ⟩
+  DynP'→DynP d = transport (sym (λ i -> ⟨ unfold-DynP i ⟩)) d
+
+  DynP→DynP' : ⟨ DynP ⟩ -> ⟨ DynP' (next DynP) ⟩
+  DynP→DynP' d = transport (λ i → ⟨ unfold-DynP i ⟩) d
+
+
+  DynP-rel : ∀ d1 d2 ->
+    rel (DynP' (next DynP)) d1 d2 ->
+    rel DynP (DynP'→DynP d1) (DynP'→DynP d2)
+  DynP-rel d1 d2 d1≤d2 = transport
+    (λ i → rel (unfold-DynP (~ i))
+      (transport-filler (λ j -> ⟨ unfold-DynP (~ j) ⟩) d1 i)
+      (transport-filler (λ j -> ⟨ unfold-DynP (~ j) ⟩) d2 i))
+  d1≤d2
+
+  DynP'-rel : ∀ d1 d2 ->
+    rel DynP d1 d2 ->
+    rel (DynP' (next DynP)) (DynP→DynP' d1) (DynP→DynP' d2)
+  DynP'-rel d1 d2 d1≤d2 = transport
+    (λ i → rel (unfold-DynP i)
+      (transport-filler (λ j -> ⟨ unfold-DynP j ⟩) d1 i)
+      (transport-filler (λ j -> ⟨ unfold-DynP j ⟩) d2 i))
+    d1≤d2
+  
+
+-}
+