---
title: "Recovering formulas from codes"
module: L.Coding.FormulaRecovery
lang: en
site: "Bedrock"
description: "Recovering formulas from codes"
stage: "Internal coding: tables and uniform satisfaction"
reading_order: 55
canonical: https://bedrock.institute/en/L.Coding.FormulaRecovery.html
html: L.Coding.FormulaRecovery.html
agda_source: https://github.com/BedrockInstitute/Bedrock/blob/main/src/L/Coding/FormulaRecovery.lagda.md
prerequisites: [Base.Prelude, FOL.ZFStructure, FOL.Syntax, FOL.Manipulation.ConstantMapping, FOL.Absoluteness, V.Hierarchy, V.Coding, L.Constructible, L.Rank, L.Axioms.Numerals, L.Coding.Model, L.Coding.Closure, L.Coding.Descent, L.Coding.CodeShape]
routes: [internal-satisfaction]
translations: [https://bedrock.institute/zh/L.Coding.FormulaRecovery.md, https://bedrock.institute/ja/L.Coding.FormulaRecovery.md]
agent_guide: /llms.txt
license: CC-BY-NC-SA-4.0
---
# Recovering formulas from codes

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.

```agda
{-# OPTIONS --cubical --safe --guardedness #-}

open import Base.Prelude

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

open import FOL.ZFStructure using ( module hPropStructure )
open import FOL.Syntax
  using ( Term; Formula; _∈̇_; _≐_; _∧̇_; _∨̇_; _⇒̇_; ⊥̇; ∃̇_; ∀̇_; ∀̇∈; ∃̇∈ )
open import FOL.Manipulation.ConstantMapping using ( mapTm; mapFo )
import FOL.Absoluteness
open import V.Hierarchy {ℓ} using ( 𝒮ᵥ; ∈-induction )
open import V.Coding {ℓ} using ( pr; pr-inj; module VCode )
open import L.Constructible {ℓ} using ( 𝒮ʟ; isL; isL-trans )
open import L.Rank {ℓ} using ( rank )
open import L.Axioms.Numerals {ℓ} using ( numeralL; numeralL-fst )
open import L.Coding.Model {ℓ} using ( prʟ; prʟ-fst )
open import L.Coding.Closure {ℓ} using ( closedAt )
open import L.Coding.Descent {ℓ} using ( payload≺; leftPart; rightPart )
open import L.Coding.CodeShape {ℓ}
  using ( shapedAt; isTmAt-decode; Onto; BinWit; UnWit; bothTm; zeroPay
        ; module Peel )

import Cubical.Data.Sum as Sum
import Cubical.HITs.PropositionalTruncation as PT
open PT using ( ∥_∥₁; ∣_∣₁; squash₁ )
open import Cubical.HITs.CumulativeHierarchy.Base using ( V; _∈_ )
open import Cubical.HITs.CumulativeHierarchy.Constructions
  using ( module InfinitySet )
open InfinitySet using ( #_; sucV )

open hPropStructure 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.

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

keyOf-fst : (n : ℕ) (x : S) → fst (keyOf n x) ≡ pr (# n) (fst x)
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 ⌝ ≡ fst x) ∥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.

```agda
module Decode {K : Type ℓ} (f : K → V ℓ)
              {m : ℕ} (C A : Fin m) (γ : 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 (fst x)) n x refl
    where
    P : V ℓ → Type (ℓ-suc ℓ)
    P r = (j : ℕ) (z : S) → rank (fst z) ≡ 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 = PT.rec squash₁ fill (peel (keyOf j z) wz)
      where
      D = fst (lookup C γ)

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

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

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

      inD : (i : ℕ) (N u : S) → # i ≡ fst N → ⟨ pr (fst N) (fst u) ∈ 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 ≡ fst N
           → ⟨ pr (sucV (fst N)) (fst u) ∈ 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.

```agda
      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))))) =
        PT.rec squash₁
          (λ { (t , qt) → PT.map
            (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 (fst a) (fst b))) 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))))) =
        PT.rec squash₁
          (λ { (φ , qφ) → PT.map
            (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 (fst b) ∈ rank w ⟩) (sym qx)
                       (rightPart (# k) (fst a) (fst b)))
                     (inD j N b qN hb)) })
          (rec j a (subst (λ w → ⟨ rank (fst a) ∈ rank w ⟩) (sym qx)
                     (leftPart (# k) (fst a) (fst b)))
                   (inD j N a qN ha))
        where
        sp = split N (pr (# k) (pr (fst a) (fst b))) 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))) = PT.map
        (λ { (φ , qφ) → op φ
           , ( qop φ ∙ cong (VCode.mkTag k) qφ ∙ sym qx ) })
        (rec j a (subst (λ w → ⟨ rank (fst a) ∈ rank w ⟩) (sym qx)
                   (payload≺ (# k) (fst a)))
                 (inD j N a qN ha))
        where
        sp = split N (pr (# k) (fst a)) 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) (fst a)) 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))) = PT.map
        (λ { (φ , qφ) → op φ
           , ( qop φ ∙ cong (VCode.mkTag k) qφ ∙ sym qx ) })
        (rec (suc j) a
          (subst (λ w → ⟨ rank (fst a) ∈ rank w ⟩) (sym qx)
            (payload≺ (# k) (fst a)))
          (inD⁺ j N a qN ha))
        where
        sp = split N (pr (# k) (fst a)) 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))))) =
        PT.rec squash₁
          (λ { (t , qt) → PT.map
            (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 (fst b) ∈ rank w ⟩) (sym qx)
                             (rightPart (# k) (fst a) (fst b)))
                           (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 (fst a) (fst b))) e
        qN = sp .fst
        qx = sp .snd

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