コードから論理式を復元する

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読書案内 · 依存マップ

正しい形を持ち部分コードについて閉じたキーの集合には、実際の論理式のコードが含まれるはずです。本章では、自然数のアリティを指定したキーが、選んだ台に定数を持つ論理式を符号化することを示します。コードのランクについて帰納法を行い、その論理式の単なる存在を得ます。

英語原文

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.

英語原文

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.

英語原文

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.

英語原文

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.

英語原文

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.

{-# 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 _⊨_ )

キーと復号の主張

キーはアリティとコードの対です。復号では、アルファベット K 上でそのアリティを持ち、定数を階層へ写すと指定したコードになる論理式を求めます。命題的切り詰めによって、代表を選ぶことなくその存在を記録します。

英語原文

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) → 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

コードのランクに関する帰納法

閉包条件により直接の部分式のキーが得られ、そのランクは真に小さくなります。したがってランクに関する帰納法で部分式を復号した後、元の論理式を再構成できます。帰納命題ではアリティを変えられるようにし、量化子も同じ議論で扱います。

英語原文

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 : 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
英語原文

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

      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
英語原文

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))))) =
        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))))))))))