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Fix a universe level ℓ. Keeping the level as a parameter lets the constructions be instantiated at each required size without identifying distinct universes.

module L.Coding.FormulaRecovery {ℓ : Level} where

A collection of well-shaped keys closed under subcodes should contain genuine formula codes. Given a key with a specified natural-number arity, this chapter proves that it codes a formula whose constants come from the chosen carrier. The proof uses induction on the rank of the code and returns the mere existence of the recovered formula.

Which alphabet the formula is over is the whole point of the chapter, and it is decided here rather than at the end. A formula over the model would be recovered by the same six frames and would be useless to the consumer, whose index type is the formulas over one carrier. So the target is stated over an alphabet, the alphabet is a parameter, and the one thing the shape predicate cannot supply about it, that the carrier's members are the alphabet's image, is a hypothesis beside it.

The target being the alphabet's own coding also makes the frames shorter rather than longer. Over the model each frame had to bridge two codings, the model's and the hierarchy's, before it could compare a code with a payload; over the alphabet the code already is an element of the hierarchy and the bridge is gone.

What is not proved here is that every member is such a key, and the set is what owes it. Shapedness binds the arity component existentially and puts no condition on it, so a set holding a pair whose first component is not a numeral satisfies both halves and this theorem says nothing about it. The set the next chapter builds pins the arity from outside, by separating inside a family indexed at one fixed arity, which is why the predicate is not asked to.

The recursion runs on the rank of the code, not on the code and not on the key. Not on the code because membership does not descend into a Kuratowski pair; not on the key because the key carries the arity beside the code and rank arithmetic on a pair is a fact nobody has proved. Carrying the arity as a natural number alongside, and descending on the code alone, needs neither.

One step is peel, one descent is the previous chapter, and the ten cases collapse to six, because the ten tags have six shapes between them and what changes inside a shape is a tag and a constructor.

open import Cubical.HITs.CumulativeHierarchy.Base using ( V; _∈_ )
open import Cubical.HITs.CumulativeHierarchy.Constructions
  using ( module InfinitySet )
open InfinitySet using ( #_; sucV )

open hPropView 𝒮ʟ

module AbsL = FOL.Absoluteness.Single 𝒮ᵥ isL isL-trans
open AbsL renaming ( _⊨ᵐ_ to _⊨_ )

Keys and the decoding statement

A key pairs an arity with a code. Decoding asks for a formula of that arity over an alphabet K, such that mapping its constants into the hierarchy gives the stated code. Propositional truncation records the existence of such a formula without selecting a representative.

The code is taken of the formula's image in the hierarchy, which is what mapFo is doing there. It is not a step of the construction: relabelling commutes with every constructor definitionally, so a formula over the alphabet and its image code together exactly as a formula over the model does.

keyOf : ℕ → S → S
keyOf n x = prʟ (numeralL n) x

keyOf-fst : (n : ℕ) (x : S) → (keyOf n x) .fst ≡ pr (# n) (x .fst)
keyOf-fst n x = prʟ-fst (numeralL n) x ∙ cong₂ pr (numeralL-fst n) refl

Coded : {K : Type ℓ} (f : K → V ℓ) → ℕ → S → Type (ℓ-suc ℓ)
Coded {K} f n x = ∥ Σ[ φ ∶ Formula K n ] (VCode.⌜ mapFo f φ ⌝ ≡ x .fst) ∥₁

Induction on the rank of a code

Closure provides the keys of immediate subformulas, and their ranks are strictly smaller. Rank induction can therefore decode them before rebuilding the whole formula. The induction statement allows the arity to vary, so the same argument also covers quantifiers.

The alphabet's two parameters ride outside the induction for the same reason. Only two of the six frames look at them, the two whose payload holds a term, and they look at them by handing the hypothesis straight to the term decode.

module Decode {K : Type ℓ} (f : K → V ℓ)
              {m : ℕ} (C A : Fin m) (γ : Vec S m) (onto : Onto f A γ)
              (hcl : ⟨ γ ⊨ closedAt C ⟩) (hsh : ⟨ γ ⊨ shapedAt C A ⟩) where

open Peel C A γ hcl hsh

Wf : ℕ → S → Type (ℓ-suc ℓ)
Wf n x = ⟨ keyOf n x ∈ˢ lookup C γ ⟩

recover : (n : ℕ) (x : S) → Wf n x → Coded f n x
recover n x = ∈-induction go (rank (x .fst)) n x refl
  where
  P : V ℓ → Type (ℓ-suc ℓ)
  P r = (j : ℕ) (z : S) → rank (z .fst) ≡ r → Wf j z → Coded f j z

  go : (r : V ℓ) → ((y : V ℓ) → ⟨ y ∈ r ⟩ → P y) → P r
  go r IH j z qr wz = rec₁ squash₁ fill (peel (keyOf j z) wz)
    where
    D = (lookup C γ) .fst

    rec : (i : ℕ) (u : S) → ⟨ rank (u .fst) ∈ rank (z .fst) ⟩
        → Wf i u → Coded f i u
    rec i u lt wu = IH (rank (u .fst))
      (subst (λ w → ⟨ rank (u .fst) ∈ w ⟩) qr lt) i u refl wu

The arity numeral and the payload, read out of the key's shape.

    split : (N : S) (p : V ℓ) → (keyOf j z) .fst ≡ pr (N .fst) p
          → (# j ≡ N .fst) × (z .fst ≡ p)
    split N p e = pr-inj (sym (keyOf-fst j z) ∙ e)

    inD : (i : ℕ) (N u : S) → # i ≡ N .fst → ⟨ pr (N .fst) (u .fst) ∈ D ⟩
        → Wf i u
    inD i N u qN h = subst (λ w → ⟨ w ∈ D ⟩)
      (cong₂ pr (sym qN) refl ∙ sym (keyOf-fst i u)) h

    inD⁺ : (i : ℕ) (N u : S) → # i ≡ N .fst
         → ⟨ pr (sucV (N .fst)) (u .fst) ∈ D ⟩ → Wf (suc i) u
    inD⁺ i N u qN h = subst (λ w → ⟨ w ∈ D ⟩)
      (cong₂ pr (cong sucV (sym qN)) refl ∙ sym (keyOf-fst (suc i) u)) h

The six frames. Each takes the constructor's coding equation rather than leaving the elaborator to find it: with the constructor a variable, nothing reduces, and the unification is the whole cost. Over the alphabet the equation is still refl at every call site, because relabelling commutes with every constructor definitionally.

    atom : (k : ℕ) (op : ∀ {i} → Term K i → Term K i → Formula K i)
         → (∀ {i} (t u : Term K i)
            → VCode.⌜ mapFo f (op t u) ⌝
              ≡ VCode.mkTag k (pr VCode.⌜ mapTm f t ⌝ᵗ VCode.⌜ mapTm f u ⌝ᵗ))
         → BinWit k (bothTm A) γ (keyOf j z) → Coded f j z
    atom k op qop (N , (a , (b , (e , (ha , hb))))) =
      rec₁ squash₁
        (λ { (t , qt) → map₁
          (λ { (u , qu) → op t u
             , ( qop t u
               ∙ cong (VCode.mkTag k) (cong₂ pr qt qu)
               ∙ sym qx ) })
          (isTmAt-decode f zero (suc (suc zero)) (suc (suc (suc (suc A))))
            (b ∷ a ∷ N ∷ keyOf j z ∷ γ) j (sym qN) onto hb) })
        (isTmAt-decode f (suc zero) (suc (suc zero)) (suc (suc (suc (suc A))))
          (b ∷ a ∷ N ∷ keyOf j z ∷ γ) j (sym qN) onto ha)
      where
      sp = split N (pr (# k) (pr (a .fst) (b .fst))) e
      qN = sp .fst
      qx = sp .snd

    binSame : (k : ℕ) (op : ∀ {i} → Formula K i → Formula K i → Formula K i)
            → (∀ {i} (φ ψ : Formula K i)
               → VCode.⌜ mapFo f (op φ ψ) ⌝
                 ≡ VCode.mkTag k (pr VCode.⌜ mapFo f φ ⌝ VCode.⌜ mapFo f ψ ⌝))
            → BinSame k (keyOf j z) → Coded f j z
    binSame k op qop (N , (a , (b , (e , (ha , hb))))) =
      rec₁ squash₁
        (λ { (φ , qφ) → map₁
          (λ { (ψ , qψ) → op φ ψ
             , ( qop φ ψ
               ∙ cong (VCode.mkTag k) (cong₂ pr qφ qψ)
               ∙ sym qx ) })
          (rec j b (subst (λ w → ⟨ rank (b .fst) ∈ rank w ⟩) (sym qx)
                     (rightPart (# k) (a .fst) (b .fst)))
                   (inD j N b qN hb)) })
        (rec j a (subst (λ w → ⟨ rank (a .fst) ∈ rank w ⟩) (sym qx)
                   (leftPart (# k) (a .fst) (b .fst)))
                 (inD j N a qN ha))
      where
      sp = split N (pr (# k) (pr (a .fst) (b .fst))) e
      qN = sp .fst
      qx = sp .snd

    unSame : (k : ℕ) (op : ∀ {i} → Formula K i → Formula K i)
           → (∀ {i} (φ : Formula K i)
              → VCode.⌜ mapFo f (op φ) ⌝ ≡ VCode.mkTag k VCode.⌜ mapFo f φ ⌝)
           → UnSame k (keyOf j z) → Coded f j z
    unSame k op qop (N , (a , (e , ha))) = map₁
      (λ { (φ , qφ) → op φ
         , ( qop φ ∙ cong (VCode.mkTag k) qφ ∙ sym qx ) })
      (rec j a (subst (λ w → ⟨ rank (a .fst) ∈ rank w ⟩) (sym qx)
                 (payload≺ (# k) (a .fst)))
               (inD j N a qN ha))
      where
      sp = split N (pr (# k) (a .fst)) e
      qN = sp .fst
      qx = sp .snd

    konst : (k : ℕ) (op : ∀ {i} → Formula K i)
          → (∀ i → VCode.⌜ mapFo f (op {i}) ⌝ ≡ VCode.mkTag k (# 0))
          → UnWit k zeroPay γ (keyOf j z) → Coded f j z
    konst k op qop (N , (a , (e , ha))) = ∣ op
      , ( qop j ∙ cong (VCode.mkTag k) (sym (ha ∙ numeralL-fst 0))
        ∙ sym qx ) ∣₁
      where
      qx = split N (pr (# k) (a .fst)) e .snd

    unSucc : (k : ℕ) (op : ∀ {i} → Formula K (suc i) → Formula K i)
           → (∀ {i} (φ : Formula K (suc i))
              → VCode.⌜ mapFo f (op φ) ⌝ ≡ VCode.mkTag k VCode.⌜ mapFo f φ ⌝)
           → UnSucc k (keyOf j z) → Coded f j z
    unSucc k op qop (N , (a , (e , ha))) = map₁
      (λ { (φ , qφ) → op φ
         , ( qop φ ∙ cong (VCode.mkTag k) qφ ∙ sym qx ) })
      (rec (suc j) a
        (subst (λ w → ⟨ rank (a .fst) ∈ rank w ⟩) (sym qx)
          (payload≺ (# k) (a .fst)))
        (inD⁺ j N a qN ha))
      where
      sp = split N (pr (# k) (a .fst)) e
      qN = sp .fst
      qx = sp .snd

    bnd : (k : ℕ) (op : ∀ {i} → Term K i → Formula K (suc i) → Formula K i)
        → (∀ {i} (t : Term K i) (φ : Formula K (suc i))
           → VCode.⌜ mapFo f (op t φ) ⌝
             ≡ VCode.mkTag k (pr VCode.⌜ mapTm f t ⌝ᵗ VCode.⌜ mapFo f φ ⌝))
        → BinSucc k (keyOf j z) → Coded f j z
    bnd k op qop (N , (a , (b , (e , (ha , hb))))) =
      rec₁ squash₁
        (λ { (t , qt) → map₁
          (λ { (φ , qφ) → op t φ
             , ( qop t φ
               ∙ cong (VCode.mkTag k) (cong₂ pr qt qφ)
               ∙ sym qx ) })
          (rec (suc j) b (subst (λ w → ⟨ rank (b .fst) ∈ rank w ⟩) (sym qx)
                           (rightPart (# k) (a .fst) (b .fst)))
                         (inD⁺ j N b qN hb)) })
        (isTmAt-decode f zero (suc zero) (suc (suc (suc A)))
          (a ∷ N ∷ keyOf j z ∷ γ) j (sym qN) onto ha)
      where
      sp = split N (pr (# k) (pr (a .fst) (b .fst))) e
      qN = sp .fst
      qx = sp .snd

    fill : PeelWit (keyOf j z) → Coded f j z
    fill =
      ⊎-rec (atom 0 _∈̇_ (λ _ _ → refl))
      (⊎-rec (atom 1 _≐_ (λ _ _ → refl))
      (⊎-rec (binSame 2 _∧̇_ (λ _ _ → refl))
      (⊎-rec (binSame 3 _∨̇_ (λ _ _ → refl))
      (⊎-rec (binSame 4 _⇒̇_ (λ _ _ → refl))
      (⊎-rec (konst 5 ⊥̇ (λ _ → refl))
      (⊎-rec (unSucc 6 ∃̇_ (λ _ → refl))
      (⊎-rec (unSucc 7 ∀̇_ (λ _ → refl))
      (⊎-rec (bnd 8 ∀̇∈ (λ _ _ → refl))
      (bnd 9 ∃̇∈ (λ _ _ → refl))))))))))