Primitive Recursive Bars Are Inductive

Using Martín Escardó’s « effectful forcing » technique, we demonstrate the constructive validity of Brouwer’s monotone Bar Theorem for any System T-definable bar. We have not assumed any non-constructive (Classical or Brouwerian) principles in this proof, and have carried out the entire development formally in the Agda proof assistant [13] for Martin-Löf ’s Constructive TypeTheory. In 2013, Martín Escardó pioneered a technique called « effectful forcing » for demonstrating non-constructive (Brouwerian) principles for the definable functionals of Gödel’s SystemT [7], including the continuity of functionals on the Baire space and uniform continuity of functionals on the Cantor space. Effectful forcing is a remarkably simpler alternative to standard sheaf-theoretic forcing arguments, using ideas from programming languages, including computational effects, monads and logical relations. Following a suggestion from Thierry Coquand [2, 3], the author learnt that Brouwer’s controversial BarTheoremcould in principle be validated in a Bethmodel by instantiating the premise of barhood at a « generic point », which would yield an inductive mental construction of barhood. In this paper, we put an analogous version of this idea into practice using Escardó’s method. 1 Brouwer’s BarThesis There are many versions of the Bar Thesis and its corollary, the bar induction principle, but we will describe here a particularly perspicuous one. First we will define a point-free notion of topological space called a « spread ». 1 ar X iv :1 60 8. 03 81 4v 2 [ m at h. L O ] 1 2 Se p 20 16 Definition 1.1. A spread consists in a set X and a species S of finite sequences (nodes) ?⃗? ∈ X⋆ such that the following hold: ⟨⟩ ∈ S ?⃗? ∈ S ∃x ∈ X. ?⃗?⌢x ∈ S ?⃗? ∈ S ⃗ v ≼ ?⃗? ⃗ v ∈ S Viewed as a topological space, the admitted finite sequences are the spread’s open sets (neighborhoods), and its points are the infinite sequences α ∈ XN whose every prefix ?⃗? ≺ α is admitted. The topology of a spread is given by the notion of a « bar » or « cover ». We say that a species of nodes Q ⊆ S covers (bars) a node ?⃗? when every infinite sequence out of ?⃗? has a prefix inQ. Formally: ∀α ≻ ?⃗?. ∃k ∈ N. α [k] ∈ Q ?⃗? ◁ Q covering A speciesQ is calledmonotone when, if ?⃗? ∈ Q, we also have ?⃗?⌢x ∈ Q for any x ∈ X. Inductive Covers Separately, we define an inductive version of the covering relation for a monotone species of nodes Q, defined as the least relation closed under the following two rules of inference: ?⃗? ∈ Q ?⃗? ◀ Q η ∀x ∈ X. ?⃗? ⌢x ◀ Q ?⃗? ◀ Q ϝ Theorem1.2. AssumingQ is monotone, Brouwer’s contested ζ inference is admissible: ?⃗? ◀ Q ?⃗?⌢x ◀ Q ζ Proof. By case on the premise. 1. If the premise was η, then by monotonicity ofQ and η. 2. If the premise was ϝ, then we have for any y ∈ X, ?⃗?⌢y ◀ Q. Choose y ≡ x. Theorem 1.3. The inductive relation ?⃗? ◀ Q is a sound characterization of covering, i.e. the following rule of inference is justified: ?⃗? ◀ Q ?⃗? ◁ Q soundness