Coq as a Metatheory for Nuprl with Bar Induction

These past few years, we have been experimenting in Nuprl with versions of Brouwer’s Bar Induction principle. Until recently we had no formal proof that these rules are valid Nuprl reasoning principles. Thanks to our formalization of Nuprl’s metatheory in Coq, we can now rigorously check whether these principles are consistent with Nuprl. In this paper we present a proof, using our Coq framework, of the validity of Brouwer’s Bar Induction principle on sequences of natural numbers. To prove this result we added all Coq functions from natural numbers to natural numbers to Nuprl’s computation system. Introduction. Nuprl [9, 3] is a dependent type theory à la Martin-Löf [15] based on an untyped functional programming language. Nuprl has a rich type theory including identity (or equality) types, a hierarchy of universes, W types, quotient types [10], set types, union and (dependent) intersection types [14], image types [16], PER types [4], simulation types [18], and partial types [11]. Type checking is undecidable but in practice this is mitigated by type inference and checking heuristics implemented as tactics. Nuprl types are defined as partial equivalence relations (PERs) on closed terms [2, 1, 11]. We have implemented Nuprl’s PER semantics in Coq and verified a large number of its inference rules [5, 6]. Bar Induction. These past few years we have been experimenting with versions of Brouwer’s Bar Induction principle [8, 19]. In [7], we showed how to build parametrized families of W types using a general form of Bar Induction. We present here our first attempt at proving that this rule is valid w.r.t. Nuprl PER semantics. The rule we have proved is: H ⊢ ↓(X 0 norm(c, 0)) BY [BarInduction] (dec) H , n : N, s : Nn ⊢ B n s ∨ ¬B n s (bar) H , s : N ⊢ ↓∃n : N. B n norm(s, n) (imp) H , n : N, s : Nn ,m : B n s ⊢ X n s (ind) H , n : N, s : Nn , x : (∀m : N. X (n+ 1) ext(s, n,m)) ⊢ X n s where norm(s, n) = λx.if x<0 then ⊥ else if x<n then s(x) else ⊥ ext(s, n,m) = λx.if x =Z n then m else s(x) ↓T = {Unit | T } The normalized sequence norm(s, n) returns s(x) for inputs x between 0 and n, and otherwise returns ⊥. Our first attempt to verify this rule did not use norm and, to use [BarInduction], required one to, in addition to the above subgoals, prove that X is a well-formed function of type n:N → Nn → Type. This is undesirable and can be avoided using norm. As mentioned in [7], we can define a bar recursion operator and using this squashed bar induction rule (using the squashing operator ↓), we can prove that bar recursion is a realizer for bar induction. We have proved that this rule is true in our impredicative metatheory (in Prop) following Dummett’s “standard” classical proof [12, pp.55]. Our Coq implementation is available at