On the Formal Constructive Theory of Computable ...

Density theorem

Theorem (HKS, 2010)
For every type ρ = ρ1 → ... → ρk → ι there is a decidable formula
TExtρ s.t.

∀U∈Conρ(U ⊆ {a : TExtρ(U, a)} ∈ Tρ).

Proof.
Informal proof within the Scott model in [HKS, 2010], and sketch
of its formalization within TCF+.

Approximation objects in TCF+

Approximation variables:
a∗ : Tok∗ρ(Tokρ)|U∗ : LTokρ∗(LTokρ)|(U, b) : LTokρ×Tokσ|W : L(LTokρ).
Approximation terms:
a.v.|Cs0∗...sn∗−1|f (U∗, a∗) : Tokρ∗(Tokρ)|g (U∗, a∗) : LTokρ∗(LTokρ).

Some basic token-valued functions

1 ∈˙ ρ : Tok∗ρ → LTok∗ρ → TokB, (a∗∈˙ ρU∗).

2 ⊆˙ ρ : LTokρ∗ → LTokρ∗ → TokB, (U∗⊆˙ ρV ∗).

3 ParallelTrueSub : LTokB → LTokρ → LTokρ, from a list

(b1 :: . . . :: bt) we get the list of all bj ’s which correspond to

the tt-values of (boolean1 :: . . . :: booleant).
−−−→
4 n,ρ : (LTokρ)n → LTokρ: concatenation of n lists of
i =1

ρ-tokens.

The formal information systems
Syntρ = (Tokρ, LTokρ, Conρ, ρ)

1 We simoultaneously define predicates Tok∗ι , Tok∗ρ→σ, LTokι
LTokρ→σ, e.g.,
LTokρ→σ (nilρ→σ ),
Tokρ→σ(U, a) → LTokρ→σ(W ) → LTokρ→σ((U :: a) :: W ),

The formal information systems
Syntρ = (Tokρ, LTokρ, Conρ, ρ)

31 We simoultaneously define predicates Tok∗ι , Tok∗ρ→σ, LTokι
LTokρ→σ, e.g.,
LTokρ→σ (nilρ→σ ),

Tokρ→σ(U, a) → LTokρ→σ(W ) → LTokρ→σ((U :: a) :: W ),

2 and the token-valued functions ι, Conι, ρ→σ,
Conρ→σ : LTokρ→σ → TokB, e.g.,

Conρ→σ(W ) =

|W |,B |W |,L(σ)

[Cˆonσ(ParallelTrueSub( Hρ,i (πl (W )), Θσ,i (πr (W ))].

B i=1 i=1

The formal information systems
Syntρ = (Tokρ, LTokρ, Conρ, ρ)

31 We simoultaneously define predicates Tok∗ι , Tokρ∗→σ, LTokι
LTokρ→σ, e.g.,
LTokρ→σ (nilρ→σ ),

Tokρ→σ(U, a) → LTokρ→σ(W ) → LTokρ→σ((U :: a) :: W ),

2 and the token-valued functions ι, Conι, ρ→σ,
Conρ→σ : LTokρ→σ → TokB, e.g.,

Conρ→σ(W ) =

|W |,B |W |,L(σ)

[Cˆonσ(ParallelTrueSub( Hρ,i (πl (W )), Θσ,i (πr (W ))].

B i=1 i=1

Theorem (P 2012)
Syntρ = (Tokρ, LTokρ, Conρ, ρ) is a formal information system.

Formulas of TCF+

13 Prime ∆-formulas: a↑b, a∈˙ ρU, U ρ a
2 ∆-formulas: Built from prime ∆-formulas by →, ∧, ∨

(bounded quantifiers ∀a∈˙ ρU , ∃a∈˙ ρU are definable). They are
decidable.
3 Prime Σ-formulas: prime ∆-formulas or of the form a ∈ρ x,
or a ∈ρ C , or a ∈ρ D where aTokρ and xρ, C ρ, Dρ.
4 Σ-formulas: are built as follows

1 A prime Σ-formula is a Σ-formula.
2 If A0 is a ∆-formula and B is a Σ-formula, then A0 → B is a

Σ-formula.
3 Σ-formulas are closed under ∧, ∨, bounded quantifiers and

existential quantifiers over token variables.

5 Prime formulas: either prime Σ-formulas or of the form
Tρ(x), x ≈ρ y .

6 Formulas: are built from prime formulas by →, ∧, ∨, ∀, ∃,
where the quantifiers are w.r.t. all kinds of variables.