Proof-relevance in Bishop-style constructive mathematics

Abstract Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory 
$(\mathrm{BST})$
 , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas 
$\phi$
 of BISH a set 
${\texttt{Prf}}(\phi)$
 of “proofs” or witnesses of 
$\phi$
 . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory 
$(\mathrm{MLTT})$
 . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.

Yet, in both books, many issues were left untouched, a fact that often was a source of confusion. In many occasions, especially in the measure theory of Bishop and Cheng (1972) and Bishop and Bridges (1985), the powerset was treated as a set, while in the measure theory of Bishop (1967), Bishop generally avoided the powerset by using appropriate families of subsets instead. In later works of Bridges and Richman, like Bridges and Richman (1987) and Mines et al. (1988), the powerset was clearly used as a set, in contrast though, to the predicative spirit of Bishop (1967).
The concept of a family of sets indexed by a (discrete) set was asked to be defined in Bishop (1967, Exercise 2, p. 72), and a definition, attributed to Richman, was given in Bishop and Bridges (1985, Exercise 2, p. 78). An elaborate study though, of this concept within BISH, was missing, despite its central character in the measure theory of Bishop (1967), its extensive use in the theory of Bishop spaces (Petrakis 2015a(Petrakis ,b, 2016a(Petrakis ,b, 2019a(Petrakis ,b, 2020a(Petrakis ,b, 2021(Petrakis , to appear, 2022a and in abstract constructive algebra (Mines et al. 1988). Actually, in Mines et al. (1988) Richman introduced the more general notion of a family of objects of a category indexed by some set, but the categorical component in the resulting mixture of Bishop's set theory and category theory was not explained in constructive terms. 2 The type-theoretic interpretation of Bishop's set theory into the theory of setoids (see especially the work of Palmgren 2005Palmgren , 2012aPalmgren , 2012bPalmgren , 2013Palmgren , 2014Palmgren , 2017Palmgren and Wilander 2014) has become nowadays the standard way to understand Bishop sets (as far as I know, this is a term due to Palmgren). A setoid is a type A in a fixed universe U equipped with a term : A → A → U that satisfies the properties of an equivalence relation. The identity type of Martin-Löf 's intensional type theory (MLTT) (see Martin-Löf 1998), expresses, in a proof-relevant way, the existence of the least reflexive relation on a type, a fact with no counterpart in Bishop's set theory. As a consequence, the free setoid on a type is definable (see Palmgren 2014, p. 90), and the presentation axiom in setoids is provable. Moreover, in MLTT, the families of types over a type I are the type I → U , which belongs to the successor universe U of U . In Bishop's set theory though, where only one universe of sets is implicitly used, the set character of the totality of all families of sets indexed by some set I is questionable from the predicative point of view (see our comment after Definition 11).

On Bishop Set Theory (BST)
Bishop set theory (BST), elaborated in Petrakis (2020c), is an informal, constructive theory of totalities and assignment routines that serves as a "completion" of Bishop's theory of sets. Its first aim is to fill in the "gaps, " or highlight the fundamental notions that were suppressed by Bishop in his account of the set theory underlying BISH. Its second aim is to serve as an intermediate step between Bishop's theory of sets and an adequate and faithful formalisation of BISH in Feferman's sense (Feferman 1979). To assure faithfulness, we use concepts or principles that appear, explicitly or implicitly, in BISH. Next we describe briefly the features of BST that "complete" Bishop's theory of sets in Petrakis (2020c).
1. Explicit use of a universe of sets. Bishop used a universe of sets only implicitly. For example, he "roughly" describes in Bishop (1967, p. 72), a set-indexed family of sets as . . . a rule which assigns to each t in a discrete set T a set λ(t).
Every other rule, or assignment routine mentioned by Bishop is from one given totality, the domain of the rule, to some other totality, its codomain. The only way to make the rule of a family of sets compatible with this pattern is to employ a totality of sets. In the unpublished manuscript (Bishop 1968), Bishop explicitly used a universe in his formulation of dependenttype theory as a formal system for BISH. Here we use the totality V 0 of sets, which is defined in an open-ended way, and it contains the primitive set N and all defined sets. V 0 itself is not a set, but a class. It is a notion instrumental to the definition of dependent operations and of a set-indexed family of sets.

Elaboration of the theory of families of sets.
With the use of the universe V 0 , of the notion of a non-dependent assignment routine λ 0 from an index-set I to V 0 , and of a certain dependent operation λ 1 , we define explicitly in Definition 11 the notion of a family of sets indexed by I. Although an I-family of sets is a certain function-like object, it can be understood also as an object of a one level higher than that of a set. The corresponding notion of a "function" from an I-family to an I-family M is that of a family-map. Operations between sets generate operations between families of sets and their family-maps. If the index-set I is a directed set, the corresponding notion of a family of sets over it is that of a direct family of sets. Families of subsets of a given set X over an index-set I are special I-families that deserve an independent treatment. Families of equivalence classes, families of partial functions, families of complemented subsets, and direct families of subsets are some of the variations of set-indexed families of subsets that are studied in Petrakis (2020c) with many applications in Bishop-style constructive mathematics.
Here we apply the general theory of families of sets, in order to reveal proof-relevance in BISH. Classical mathematics is proof-irrelevant, as it is indifferent to objects that "witness" a relation or a more complex formula. On the other extreme, Martin-Löf type theory is proof-relevant, as every element of a type A is a proof of the "proposition" A. Bishop's presentation of BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof-sets that provide a BHK-interpretation within BST of formulas that correspond to the standard atomic formulas of a first-order theory. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas φ of BISH a set Prf(φ) of "proofs" or witnesses of φ. Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional MLTT. Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality (FunExt), or with Voevodsky's axiom of univalence (UA)) can be translated into BISH. While Bishop's theory of sets is standardly understood through its translation to MLTT (see e.g., Coquand et al. 2005), the development of BST offers a partial translation in the converse direction.

Outline of this Paper
(1) In Section 4, we present the fundamental notions of BST that are used in the rest of the paper. (2) In Section 6, we define within BST the notion of a set-indexed family of sets and its corresponding -and -sets. Moreover, we provide all new set-indexed families of sets constructed from given ones that are used in the following sections. (3) In Section 7, we define the notion of a set-relevant family of sets, a generalisation of a family of sets over a set with a proof-relevant equality, introduced in Section 11. (4) In Section 9, we provide a BHK-interpretation of a large part of BISH within BST, including many motivating examples. (5) In Section 10, we present interesting totalities in BISH equipped with a proof-relevant equality. (6) In Section 11, we introduce the notion of a Martin-Löf set in BST, an abstract version of a set in BST with a proof-relevant equality, and we prove some first fundamental properties of Martin-Löf sets. (7) In Section 12, we translate results on contractible sets and subsingletons from Homotopy Type Theory into BST.

Fundamental Notions of BST
The logical framework of BST is first-order intuitionistic logic with equality (see Schwichtenberg and Wainer 2012, Chapter 1). This primitive equality between terms is denoted by s := t, and it is understood as a definitional, or logical, equality. That is, we read the equality s := t as "the term s is by definition equal to the term t." If φ is an appropriate formula, for the standard axiom for equality [a := b & φ(a)] ⇒ φ(b), we use the notation [a := b & φ(a)] :⇒ φ(b). The equivalence notation :⇔ is understood in the same way. The set (N = N , = N ) of natural numbers, where its canonical equality is given by m = N n :⇔ m := n, and its canonical inequality by m = N n :⇔ ¬(m = N n), is primitive. The standard Peano-axioms are associated to N. A global operation (·, ·) of pairing is also considered primitive. That is, if s, t are terms, their pair (s, t) is a new term. The corresponding equality axiom is (s, t) := (s , t ) :⇔ s := s & t := t . The n-tuples of given terms, for every n larger than 2, are definable. The global projection routines pr 1 (s, t) := s and pr 2 (s, t) := t are also considered primitive. The corresponding global projection routines for any n-tuples are definable.
An undefined notion of mathematical construction, or algorithm, or of finite routine is considered as primitive. The main primitive objects of BST are totalities and assignment routines. Sets are special totalities and functions are special assignment routines, where an assignment routine is a a special finite routine. All other equalities in BST are equalities on totalities defined though an equality condition. A predicate on a set X is a bounded formula P(x) with x a free variable ranging over X, where a formula is bounded, if every quantifier occurring in it is over a given set.

Definition 1. (i) A primitive set
A is a totality with a given membership x ∈ A, and a given equality x = A y, that satisfies axiomatically the properties of an equivalence relation. The set N of natural numbers is the only primitive set considered here.
where M X is a formula with x as a free variable. (iii) There is a special "open-ended" defined totality V 0 , which is called the universe of (predicative) sets. V 0 is not defined through a membership-condition, but in an open-ended way. When we say that a defined totality X is considered to be a set we "introduce" X as an element of V 0 . We do not add the corresponding induction, or elimination principle, as we want to leave open the possibility of adding new sets in V 0 .
(iv) A defined preset X, or simply, a preset, is a defined totality X the membership condition M X of which expresses a construction. No quantification over V 0 occurs in M X .
(v) A defined totality X with equality, or simply, a totality X with equality is a defined totality X equipped with an equality condition x = X y :⇔ E X (x, y), where E X (x, y) is a formula with free variables x and y that satisfies the conditions of an equivalence relation. (vi) A defined set is a preset with a given equality. (vii) A set is either a primitive set or a defined set. (viii) A totality is a class, if it is the universe V 0 , or if quantification over V 0 occurs in its membership condition.

Definition 2. A bounded formula on a set X is called an extensional property on X, if
The totality X P generated by P(x) is defined by x ∈ X P :⇔ x ∈ X & P(x), x ∈ X P :⇔ x ∈ X & P(x), and the equality of X P is inherited from the equality of X. We also write X P := {x ∈ X | P(x)}, X P is considered to be a set, and it is called the extensional subset of X generated by P.
It is expected that the proof-terms in PrfEql 0 (X, Y) are compatible with the properties of the equivalence relation X = V 0 Y. This means that we can define a distinguished proof-term refl(X) ∈ PrfEql 0 (X, X) that proves the reflexivity of It is immediate to see that these operations satisfy the groupoid laws: (i) refl(X) * (f , g) = PrfEql 0 (X,Y) (f , g) and (f , g) * refl(Y) = PrfEql 0 (X,Y) (f , g).
If g is a modulus of surjectivity for f , we also say that f is a retraction and Y is a retract of X. If y ∈ Y, the fiber fib f (y) of f at y is the is a center of contraction for fib f (y).
Definition 8. Let I be a set and λ 0 : I V 0 a nondependent assignment routine from I to V 0 . A dependent operation over λ 0 , in symbols is an assignment routine that assigns to each element i in I an element (i) in the set λ 0 (i). If i ∈ I, we call (i) the i-component of , and we also use the notation i := (i). An assignment routine is either a nondependent assignment routine or a dependent operation over some nondependent assignment routine from a set to the universe. If : i∈I λ 0 (i), := :⇔ ∀ i∈I i := i . If := , we say that and are definitionally equal. Let A(I, λ 0 ) be the totality of dependent operations over λ 0 , equipped with the canonical equality = A(I,λ 0 ) :⇔ ∀ i∈I i = λ 0 (i) i . The totality A(I, λ 0 ) is considered to be a set.
If f : X → Y, let fib f : Y V 0 be defined by y → fib f (y), for every y ∈ Y. If f is contractible, then by Definition 6 every fiber fib f (y) of f is contractible. A modulus of centers of contraction for a contractible function f is a dependent operation centre f : y∈Y fib f (y), such that centre f y := centre f (y) is a center of contraction for f .

Subsets
Definition 9. Let X be a set. A subset of X is a pair (A, i X A ), where A is a set and ı X A : A → X is an embedding. If (A, i X A ) and (B, i X B ) are subsets of X, then A is a subset of B, in symbols In this case we use the notation f : A ⊆ B. Usually we write A instead of (A, i X A ). The totality of the subsets of X is the powerset P(X) of X, and it is equipped with the equality Since the membership condition for P(X) requires quantification over V 0 , the totality P(X) is a class. Clearly, (X, id X ) ⊆ X. If X P is an extensional subset of X (see Definition 2), then (X P , i X P ) ⊆ X, where i X P : X P X is defined by i X P (x) := x, for every x ∈ X P .

Proposition 10. If A, B ⊆ X, and f , h : A ⊆ B, then f is an embedding, and f
The "internal" equality of subsets implies their "external" equality as sets, i.e., (f , g) : equipped with the canonical equality of pairs as in the case of PrfEql 0 (X, Y). Because of Proposition 10, the set PrfEql 0 (A, B) is a subsingleton, i.e.,

Set-Indexed Families of Sets
Roughly speaking, a family of sets indexed by some set I is an assignment routine λ 0 : I V 0 that behaves like a function i.e., if i = I j, then λ 0 (i) = V 0 λ 0 (j). Next follows an exact formulation of this description that reveals the witnesses of the equality λ 0 (i) = V 0 λ 0 (j).
Definition 11. If I is a set, a family of sets indexed by I, or an I-family of sets, is a pair := (λ 0 , λ 1 ), where λ 0 : I V 0 , and λ 1 , a modulus of function-likeness for λ 0 , is given by such that the transport maps λ ij of satisfy the following conditions: (a) For every i ∈ I, we have that λ ii := id λ 0 (i) . (b) If i = I j and j = I k, the following diagram commutes I is the index-set of the family . If X is a set, the constant I-family of sets X is the pair C X := (λ X 0 , λ X 1 ), where λ 0 (i) := X, for every i ∈ I, and λ 1 (i, j) := id X , for every (i, j) ∈ D(I). The pair 2 2 2 := (λ 2 2 2 0 , λ 2 2 2 1 ), where λ 2 2 2 0 : 2 2 2 V 0 with λ 2 2 2 0 (0) := X, λ 2 2 2 0 (1) := Y, and λ 2 2 2 1 (0, 0) := id X and λ 2 2 2 1 (1, 1) := id Y , is the 2 2 2-family of X and Y. The n n n-family n n n of the sets X 1 , . . . X n , where n ≥ 1, and the N-family N := (λ N 0 , λ N 1 ) of the sets (X n ) n∈N are defined similarly. Let := (λ 0 , λ 1 ) and M := (μ 0 , μ 1 ) be I-families of sets. A family-map from to M, in symbols : ⇒ M is a dependent operation : i∈I F λ 0 (i), μ 0 (i) such that for every (i, j) ∈ D(I) the following diagram commutes F λ 0 (pr 1 (z)), λ 0 (pr 2 (z)) , but, for simplicity, we avoid the use of the primitive projections pr 1 , pr 2 . Condition (a) of Definition 11 could have been written as λ ii = F(λ 0 (i),λ 0 (i)) id λ 0 (i) . If i = I j, then by conditions (b) and (a) of Definition 11 we get id λ 0 (i) := λ ii = λ ji • λ ij and id λ 0 (j) := λ jj = λ ij • λ ji , i.e., (λ ij , λ ji ) : λ 0 (i) = V 0 λ 0 (j). In this sense, λ 1 is a modulus of function-likeness for λ 0 . It is natural to accept the totality Map( , M) as a set. If Fam(I) was a set though, the constant I-family with value Fam(I) would be defined though a totality in which it belongs to. From a predicative point of view, this cannot be accepted. The membership condition of the totality Fam(I) though does not depend on the universe V 0 , therefore it is also natural not to consider Fam(I) to be a class. Hence, Fam(I) is a totality "between" a (predicative) set and a class. For this reason, we say that Fam(I) is an impredicative set.
By the definition of the canonical equality on i∈I λ 0 (i), we get that pr 1 is a function.
Definition 15. Let := (λ 0 , λ 1 ) be an I-family of sets. The -indexing of is the pair Clearly, is a family of sets over i∈I λ 0 (i).
Definition 16. Let := (λ 0 , λ 1 ) be an I-family of sets. The second projection on i∈I λ 0 (i) is the dependent operation pr 2 : (i,x)∈ i∈I λ 0 (i) λ 0 (i), defined by pr 2 (i, x) := pr 2 (i, x) := x, for every (i, x) ∈ i∈I λ 0 (i). We write pr 2 , when the family of sets is clearly understood from the context. Definition 17. Let := (λ 0 , λ 1 ) be an I-family of sets. The totality i∈I λ 0 (i) of dependent functions over , or the -set of , is defined by and it is equipped with the canonical equality and the canonical inequality of the set A(I, λ 0 ). If X is a set and X is the constant I-family X (see Definition 11), we use the notation Remark 18. If := (λ 0 , λ 1 ) is an I-family of sets and := (σ 0 , σ 1 ) is the -indexing of , then pr 2 is a dependent function over .
Proof. By Definition 16 the second projection pr 2 of is the dependent assignment Next we define new families of sets generated by a given family of sets indexed by the product X × Y of X and Y.

Set-Relevant Families of Sets
In general, we may want to have more than one transport maps from λ 0 (i) to λ 0 (j), if i = I j. In this case, to each (i, j) ∈ D(I) we associate a set of transport maps.

Definition 20. If I is a set, a set-relevant family of sets indexed by I, is a triplet
such that the following conditions hold: It is immediate to show that if := (λ 0 , λ 1 ) ∈ Fam(I), then generates a set-relevant family over I, where ε λ 0 (i, j) := 1 1 1, and λ 2 (i, j), 0) := λ ij , for every (i, j) ∈ D(I).
A motivation for the definitions of * i∈I λ 0 (i) and * i∈I λ 0 (i) is provided, respectively, by

Set-Indexed Families of Subsets
Roughly speaking, a family of subsets of a set X indexed by some set I is an assignment routine λ 0 : I P(X) that behaves like a function, i.e., if i = I j, then λ 0 (i) = P(X) λ 0 (j). The following definition is a formulation of this rough description that reveals the witnesses of the equality λ 0 (i) = P(X) λ 0 (j). This is done "internally, " through the embeddings of the subsets into X. The equality λ 0 (i) = V 0 λ 0 (j), which in the previous chapter is defined "externally" through the transport maps, follows, and a family of subsets is also a family of sets.

Definition 23. Let X and I be sets. A family of subsets of X indexed by I, or an I-family of subsets of X, is a triplet
such that the following conditions hold: (a) For every i ∈ I, the function E X i : λ 0 (i) → X is an embedding. (b) For every i ∈ I, we have that λ ii := id λ 0 (i) .
is a modulus of embeddings for λ 0 , and λ 1 a modulus of transport maps for λ 0 . Let := (λ 0 , λ 1 ) be the I-family of sets that corresponds to (X).
Proposition 24. Let X and I be sets, λ 0 : I V 0 , E X a modulus of embeddings for λ 0 , and λ 1 a modulus of transport maps for λ 0 . The following are equivalent.
if and only if the above triangles are commutative. The implication (ii)⇒(i) follows immediately from the equivalence between the commutativity of the above pairs of diagrams.

Definition 25. Let X be a set and (A, i X
is the 2 2 2-family of subsets A and B of X. The n-family n (X) of the subsets (A 1 , i 1 ), . . . , (A n , i n ) of X, and the N-family of subsets (A n , i n ) n∈N of X are defined similarly. We see no obvious reason, like the one for Fam(I), not to consider Fam(I, X) to be a set. In the case of Fam(I), the constant I-family Fam(I) would be in Fam(I), while the constant I-family Fam(I, X) is not clear how could be seen as a family of subsets of X. If ν 0 (i) := Fam(I, X), for every i ∈ I, we need to define a modulus of embeddings N X i : Fam(I, X) → X, for every i ∈ I. From the given data one could define the assignment routine N Even in that case, the assignment routine N X i cannot be shown to satisfy the expected properties. Clearly, if N X i was defined by the rule N X i (X) := x 0 ∈ X, then it cannot be an embedding.
Proof. (i) By the commutativity of the following inner diagrams we get the required commutativity of the above outer diagram. If x ∈ λ 0 (i), then Because of Proposition 28(ii) all the elements of PrfEql 0 ( (X), M(X)) are equal to each other, hence the groupoid properties (i)-(iv) for PrfEql 0 ( (X), M(X)) hold trivially. Of course,

On the BHK-interpretation of BISH within BST
In the next naive definition of the BHK-interpretation of BISH, the notion of "proof " is not understood in the proof-theoretic sense. Although we agree with Streicher in Streicher (2018) that the term "witness" is better, we use the symbol Prf(φ) for traditional reasons. We could have used the symbol Wtn(φ) instead. We choose not to reduce the rule for φ ∨ ψ to the other ones, as for example is done in Beeson (1981, p. 156). The rule for ¬φ is usually reduced to the rule for implication.
Definition 29. (Naive BHK-interpretation of BISH). Let φ, ψ be formulas in BISH, such that it is understood what it means "q is a proof (or witness, or evidence) of φ" and "r is a proof of ψ." The notions of "rule" in the clauses for ( ⇒ ) and (∀) are unclear. The nature of a proof or a witness is also unclear. The interpretation of atomic formulas is also not included. In Aczel and Rathjen (2010, p. 12), the following criticism to the naive BHK-interpretation is given: Many objections can be raised against the above definition. The explanations offered for implication and universal quantification are notoriously imprecise because the notion of function (or rule) is left unexplained. Another problem is that the notions of set and set membership are in need of clarification. But in practice, these rules suffice to codify the arguments that mathematicians want to call constructive. Note also that the above interpretation (except for ⊥) does not address the case of atomic formulas.
A formal version of the above naive BHK-interpretation of BISH is a corresponding realisability interpretation (see Section 13). Following Feferman (1979), Beeson declared in Beeson (1981, p. 158) that "the fundamental relation in constructive set theory is not membership but membership-with-evidence" (MwE). All examples given by Feferman are certain extensional subsets of some set X. In MLTT, this kind of (MwE) is captured by the type x:A P(x), where P : A → U is a family of types over A : U . Here we explain how we can talk about (MwE) for extensional subsets of some set X within BST, showing that BISH, as MLTT, is capable of expressing (MwE). As all such examples known to us are extensional subsets, we do not consider the notion of a completely presented set X * , for every set X, as it is done in the formal systems T * 0 of Feferman in Feferman (1979), and in Beeson's system, found in Beeson (1981). In the system of Beeson (1981), proof-relevance is even more stressed, as to any formula φ a formula Prf φ (p) is associated by a certain rule, expressing that "p proves formula φ." The resulting formal set theory though, is, in our opinion, not attractive. The problem of the totality of proofs being a definite preset, hence the problem of quantifying over it (see Beeson 1981, p. 177) is solved by our "internal" treatment of MwE within BST. Consequently, questionable principles, like Beeson's "(MwE) is self-evident" (see Beeson 1981, p. 159), are avoided.
Proposition 30. (Membership-with-Evidence I (MwE-I)). Let X, Y be sets, and let P(x) be a property on X of the form for every x, x ∈ X and every p, p ∈ Y. Let PrfMemb P 0 : X V 0 , defined by PrfMemb P 0 (x) := {p ∈ Y | Q(x, p)}, for every x ∈ X, and let PrfMemb P Proof. (i) Let x = X x and p ∈ Y such that Q(x, p). Since p = Y p, by the extensionality of Q we get Q(x , p), and hence P(x ).
(ii) First we show that the dependent operation PrfMemb P 1 is well defined. If x = X x and p ∈ PrfMemb P 0 (x), i.e., Q(x, p), by the extensionality of Q, we get Q(x , p). Clearly, the operation PrfMemb P xx is a function. As PrfMemb P xx is given by the identity map rule, the properties of a family of sets for PrfMemb P 1 are trivially satisfied.
Actually, PrfMemb P can be seen as a family of subsets of Y over X, but now we want to work externally, and not internally. 3 For the previous result, it suffices to suppose that Q is X- , p), for every x, x ∈ X and every p ∈ Y. Notice that the extensionality of P alone does not imply neither the X-extensionality of Q nor the extensionality of Q, and it is not enough to define a function from PrfMemb P 0 (x) to PrfMemb P 0 (x ). If X P is the extensional subset of X generated by P, we write p : x ∈ X P :⇔ Q(x, p). The following proposition follows immediately from (MwE-I).
Proposition 31. (Membership-with-Evidence II (MwE-II)). Let X, Y, Z be sets, and let R(x) be a property on X of the form for every x, x ∈ X, p, p ∈ Y, and every q, q ∈ Y. Let PrfMemb R 0 : X V 0 , defined by the rule Again, PrfMemb R can be seen as a family of subsets of Y over X. If X R is the extensional subset of X generated by R, we write (p, q) : x ∈ X R :⇔ Q(x, p, q).
Clearly, the schema MwE-II can be generalised to a property S(x) on X of the form , p 1 , . . . , p n ) , for some extensional property T(p 1 , . . . , p n ) on X 1 × . . . × X n . The following scheme of defining functions on extensional subsets of sets given by existential formulas is immediate to prove.
Proposition 32. Let X, Y, X , Y be sets, and let P(x) and P(x ) properties on X and X , respectively, of the form where Q(x, p) and Q (x , p ) are extensional properties on X × Y and on X × Y , respectively.
(i) Let f : X X and f : x∈X p∈PrfMemb P 0 (x) PrfMemb P 0 (f (x)). Then the operation f PP : X P X P , defined by the rule X P x → f (x) ∈ X P , is well defined. If f is a function, then f PP is a function.
(ii) Let g : X X and g : x∈X PrfMemb P 0 (g(x)). Then the operation g P : X X P , defined by the rule X x → g(x) ∈ X P , is well defined. If g is a function, then g P is a function.
The schemata MwE-I and MwE-II are useful when a mathematical concept is defined as a property on a given set, and not as an element of the set together with some extra data. For example, in Bishop and Bridges (1985, p. 38), and in Bishop (1967, p. 34), a function f : It is also mentioned that the function ω, the so-called modulus of (uniform) continuity of f is "an indispensable part of the definition of a continuous function.'. The same concept can be defined though, through a property on the set F([a, b]) = F [a, b], R , given by an existential formula, i.e., It is this kind of definition of a mathematical notion that facilitates the definition of a set of witnesses to the membership condition of an extensional subset of a set.
Example 9.1 (Convergent sequences at x ∈ R). Let X := F(N, R), Y := F(N + , N + ). If x ∈ R, let, for every (x n ) n∈N ∈ F(N, R) If C : x n n −→, we say that C is a modulus of convergence of (x n ) n∈N at x ∈ R.
By the compatibility of the operation −, the function |.|, and the relation ≤ with the equality of R, we get the extensionality of Cauchy (x n ) n∈N :⇔ ∃ C∈F(N + ,N + ) C : Cauchy (x n ) n∈N , If C : Cauchy (x n ) n∈N , we say that C is a modulus of Cauchyness for (x n ) n∈N .
The extensionality of R (x n ) n∈N , C) :⇔ Cauchy (x n ) n∈N follows as above. By Proposition 30 PrfMemb Cauchy := PrfMemb  Similar PrfMemb-sets can be defined for the set C( [a, b]) of (uniformly) continuous real-valued functions on a compact interval [a, b] and for the set D ([a, b]) of (uniformly) differentiable functions on a compact interval [a, b]. In this framework, the Riemann-integral is not a mapping for every ω f , ω f ∈ PrfMemb Cont(f ) 0 , but it is not the accurate writing of a function from C( [a, b]) to R, only a notational convention compatible with the classical one. The following obvious generalisation (MwE-III) of (MwE-II) to relations an a set given by an existential formula is shown similarly. A variation of (MwE-III) concerns relations on finitely many different sets.

where Q(x, y, p) is an extensional property on X
The "extension" of the BHK-interpretation to what usually corresponds to atomic formulas like the equality formulas is the first part of the following definition.

Definition 34 (BHK-interpretation of BISH in BST -Part I).
Let membership conditions x ∈ X P and x ∈ X R as e.g., in Propositions 30 and 31, respectively. We define

Let a relation S(x, y) on a set X, as, e.g., in Proposition 33. We define
Prf S(x, y) := PrfRel S 0 (x, y). Let φ, ψ be formulas in BISH such that Prf(φ) and Prf(ψ) are already defined. We define

) is a formula that does not express a membership condition or a relation, are defined by
Due to the definition of the coproduct in Definition 13, the Prf-sets for ∃ x∈X φ(x) and for ∀ x∈X φ(x) are generalizations of Prf-sets for φ ∨ ψ and for φ & ψ, respectively.
Example 9.4. Let the fact: if (x n ) n∈N + ∈ F(N + , R) and x 0 ∈ R, then If χ(x n , x 0 ) is the above implication, then χ(x n , x 0 ) of the form φ(x n , x 0 ) ⇒ ψ(x n ). Its proof (see Bishop and Bridges 1985, p. 29) can be seen as a rule that sends a modulus of convergence C : x n n −→ x 0 of (x n ) n∈N + at x 0 to a modulus of Cauchyness D : Cauchy (x n ) n∈N + for (x n ) n∈N + , where D(k) := C(2k), for every k ∈ N + . This operation from PrfMemb Example 9.5. Let the fact: if x 0 ∈ R, then The formula corresponding to this proposition is To determine the Prf-set of χ * (x 0 ), we need to determine first a family of Prf-sets over F(N + , R). Prf χ(x n , x 0 ) . Example 9.6. Let the fact: if (x n ) n∈N + ∈ F(N + , R), then

Using Definition 12(ii), let
The formula corresponding to this proposition is Its proof generates a rule that associates to every C : Cauchy (x n ) n∈N + a pair (y, D), where y ∈ R and D : x n n −→ y, and y is defined by the rule y k := x D(k) ] 2k , and D(k) := 3k ∨ C(2k), for every k ∈ N + . The use of the modulus of Cauchyness in the definition of a Cauchy sequence is responsible for the avoidance of choice in the proof. Clearly, the rule C → (y, D) of the proof of θ(x n ) determines a function from Prf(ψ(x n )) to the Prf-set of the formula ∃ y∈R φ(x n , y). Since Prf(φ(x n , y) is already determined above, and as a corresponding family over F(N + , R) is determined in Example 9.1, then, using Definition 19(iii), from Definition 34 we get Prf(θ(x n )) := y∈R PrfMemb Conv y (x n ).
From the last two examples, we see how the schemes of defining new families of sets from given ones can be used in order to define canonical families of Prf-sets from given such families. These canonical families of Prf-sets are determined in the second part of our definition of the BHKinterpretation of BISH within BST. As we have already seen in the previous two examples, the following extension of Definition 34 refers to Definitions 12 and 19.

Definition 35 (BHK-interpretation of BISH in BST -Part II). Let X, Y be sets. Let
on X we associate in a canonical way the following families of sets over X, respectively:

Examples of Totalities with a Proof-Relevant Equality
The universe V 0 , the powerset P(X) of a set X, the impredicative set Fam(I) of families of sets indexed by I, the set Fam(I, X) of families of subsets of X indexed by I are some of the many examples of totalities studied in Petrakis (2020c) equipped with an equality defined through an existential formula. Here we describe some more motivating examples.

The Richman ordinals
The equality on the totality of Richman ordinals, as this is defined in Mines et al. (1988, pp. 24-28), behaves similarly to the equality on the powerset. Notice that the following definition of a well-founded relation is impredicative, as it requires quantification over the powerset of a set. If < is a binary relation on a set W, a subset H of W is called hereditary, if In this case, we write α ≤ β. In Mines et al. (1988, p. 28), it is shown that there is at most one injection from α to β. If Ord R is the totality (class) of Richman ordinals and α, β ∈ Ord R , we show the following.
As in the case of P(X), we define α = Ord R β :⇔ α ≤ β & β ≤ α, and Since the composition of injections is an injection, let and the groupoid properties for PrfEql 0 (α, β) hold trivially by the equality of all its elements.

The direct sum of a direct family of sets
Next we define the such that the transport maps λ ≺ ij of satisfy the following conditions: (a) For every i ∈ I, we have that λ ii := id λ 0 (i) .
(b) If i I j and j I k, the following diagram commutes where λ X 0 (i) := X, and λ ,X 1 (i, j) := id X , for every i ∈ I and (i, j) ∈ D (I).
Since in general I is not symmetric, the transport map λ ij does not necessarily have an inverse. Hence, λ 1 is only a modulus of transport for λ 0 , in the sense that it determines the transport maps of , and not necessarily a modulus of function-likeness for λ 0 . The direct sum i∈I λ 0 (i) over is the totality i∈I λ 0 (i) equipped with the equality . The totality i∈I λ 0 (i) of dependent functions over is defined by and it is equipped with the equality of A(I, λ 0 ).
Definition 39. Let (I, I ) be a poset, i.e., a preorder such that i I j & j I i ⇒ i = I j, for every i, j, ∈ I. A modulus of directedness for I is a function δ : I × I → I, such that for every i, j, k ∈ I the following conditions are satisfied: (δ 1 ) i I δ(i, j) and j I δ(i, j).
Proposition 40. Let δ be a modulus of directedness on a poset (I, I ), and let := (λ 0 , λ 1 ) be a family of sets over (I, I ).
Proof. (i) Since i I i, we use the definitional clause (δ 1 ) of a modulus of directedness. (ii) By (δ 3 ) we have that δ δ(i, j), i = I δ i, δ(j, i) . By (δ 1 ) and (δ 2 ), we get δ δ(i, j), i = I δ(i, j) and PrfEql 0 (j, y), (k, z) ⇔ l ∈ I jk & λ jl (y) = λ 0 (l) λ kl (z), we show that δ(m, l) ∈ I ik and λ iδ(m,l) (x) = λ 0 (δ(m,l) λ kδ(m,l) (z). By our hypotheses, i I m I δ(m, l) and k I l I δ(m, l). Moreover, If m ∈ PrfEql 0 (i, x), (j, y) and l ∈ PrfEql 0 (j, y), (k, z) , it is natural to define  (j, y). Hence, the equality on PrfEql 0 (i, x), (j, y) is defined as above, if i := j and x := y, and it is inherited from I otherwise. In order to make such a distinction though, we need to know that the previous equalities are possible, something which is not always the case without some further assumptions on the general equality :=. Of course, all aforementioned groupoid properties of * and −1 hold, if we define all elements of any set PrfEql 0 (i, x), (j, y) to be equal.

The set of reals
In Bishop and Bridges (1985, p. 18), the set of reals R is defined as an extensional subset of F(N + , Q). Specifically, where N + is the set of non-zero natural numbers. The equality on R is defined as follows: To prove though that x = R y is transitive, one needs the following characterization: Using countable choice, we get the equivalence If ω : N + → N + witnesses the equality x = R y, then ω ∨ id N + , where (ω ∨ id N + )(j) := ω(j) ∨ id N + (j) := max{ω(j), id N + (j)}, for every j ∈ N + , also witnesses the equality x = R y. Hence, without loss of generality, we can assume that ω ≥ id N + . We define If ω ∈ PrfEql 0 (x, y) and δ ∈ PrfEql 0 (y, z), we define for every j ∈ N + . In this case, ω * δ ∈ PrfEql 0 (x, z), since if n ≥ ω(2j) ∨ δ(2j), then |x n − z n | ≤ |x n − y n | + |y n − z n | ≤ 1 2j It is easy to see that * is associative, and it also compatible with the canonical equality of the sets PrfEql 0 (x, y), the one inherited from F(N + , N + ). The rest of the groupoid properties of * and −1 do not hold if we keep the canonical equality of the sets PrfEql 0 (x, y). In other words, the set PrfEql R 0 (x, y), equipped with its canonical equality, is not a ( − 1)-set. It becomes, if we truncate it, i.e., if we equip PrfEql R 0 (x, y) with the equality If (X, d) is a metric space, hence x = X y ⇔ d(x, y) = 0, for every x, y ∈ X, we define PrfEql 0 (x, y) := PrfEql 0 d(x, y), 0 .
If F is a set of real-valued functions on a set X, like a Bishop topology on X (see Petrakis 2015a), that separates the points of X i.e., If φ : R → R, let a dependent operation φ 1 : x,y∈R ω∈PrfEql 0 (x,y) PrfEql 0 (φ(x), φ(y).

Sets of integrable and measurable functions in Bishop-Cheng measure theory
In Bishop-Cheng measure theory (BCMT), Bishop and Cheng define the set of integrable functions of an integration space L := (X, L, ) (see Bishop and Bridges 1985, p. 222) as the totality where F(X) is the totality of real-valued partial functions on the set X, which are strongly extensional, i.e., if f (x) = R f (x ), then x = X x , for every x, x ∈ X. An element f of F(X) has a representation in L, if there is a sequence (f n ) ∞ n=1 of partial functions in L such that n∈N + |f n | < +∞, and A subset F of X is full, if there is g ∈ L 1 such that the domain of (the partial function) g is included in F. The equality on L 1 is defined in Bishop and Bridges (1985, p. 224) by Unfortunately, this presentation of L 1 within BCMT is highly problematic from a predicative point of view. The totality L 1 is defined through separation on F(X), which, because of the definition of a partial function from X to R, is a class, like P(X), and not a set (see Petrakis 2020c, Section 7.4). Moreover, the above equality f = L 1 g requires quantification over the class P(X). The impredicative character of BCMT hinders its computational content (see Petrakis 2020c Chapter 7, Zeuner 2019, and Petrakis and Zeuner 2022). Within this impredicative theory BCMT though, one can define If f , g, h ∈ L 1 , F ∈ PrfEql 0 (f , g), and G ∈ PrfEql 0 (g, h), it is natural to define since the intersection of full sets is a full set, and f |F = g |F & g |G = h |G ⇒ f |F∩G = h |F∩G . It is not hard to see that if we equip the sets PrfEql 0 (f , g) with the equality inherited from P(X), we get the same groupoid properties of * and −1 as in the case of R in the previous example. If is a completely extended (see Bishop and Bridges 1985, p. 223), and σ -finite integral on X (see Bishop and Bridges 1985, p. 269), and if p ≥ 1, the set L p is defined as follows (see Bishop and Bridges 1985, p. 315): where a partial function f : X R is measurable, if its domain dom(f ) is a full set, and it is appropriately approximated by elements of L 1 (for the exact definition see Bishop and Bridges 1985, p. 259). Similarly to L 1 , f = L p g : If is a σ -finite integral on X, the set L ∞ is defined as follows (see Bishop and Bridges 1985, p. 346): where a real-valued function defined on a full subset of X is essentially bounded relative to a σ -finite integral on X, if there are c > 0 and a full set F, such that |f | |F ≤ c (see Bishop and Bridges 1985, p. 346). The equality on L ∞ is defined as in L p , for p ≥ 1, and the corresponding sets PrfEql 0 (f , g) behave analogously. A complemented subset A := (A 1 , A 0 ) of X (see Petrakis 2020c, Section 2.8) is called integrable, if its characteristic function χ A is in L 1 , and then the measure on A is defined by μ(A) := χ A . If A is the totality of integrable sets with positive measure, = A is defined in Bishop and Bridges (1985, p. 346), by A = A B :⇔ χ A = L 1 χ B , and one can define PrfEql 0 (A, B) := PrfEql 0 (χ A , χ B ). All these totalities though are defined impredicatively.

Martin-Löf Sets
We give an abstract description of the previous examples of totalities (sets) with a proof-relevant equality. The introduced Martin-Löf sets give us the opportunity to transfer results and concepts from MLTT or HoTT into BST. So far, only the transition of results and concepts from BISH to MLTT was considered. This aspect of Martin-Löf sets is one of the major reasons behind their study in this paper Definition 41. Let Y be a set, and (X, = X ) a set with an equality condition of the form where θ xx (p) :⇔ p : x = X x is an extensional property on Y. Let also the nondependent assignment routine We call the structure X := (X, = X , PrfEql X 0 , refl X , −1 X , * X ) a set with a proof-relevant equality. If X is clear from the context, we may omit the subscript X from the above dependent operations. We call X a Martin-Löf set, if the following conditions hold: (ML 1 ) refl x * p = PrfEql X 0 (x,x ) p and p * refl y = PrfEql X 0 (x,x ) p, for every p ∈ PrfEql X 0 (x, x ). (ML 2 ) p * p −1 = PrfEql X 0 (x,x) refl x and p −1 * p = PrfEql X 0 (y,y) refl y , for every p ∈ PrfEql X 0 (x, x ). (ML 3 ) (p * q) * r = PrfEql X 0 (x,x ) p * (q * r), for every p ∈ PrfEql X 0 (x, x ), q ∈ PrfEql X 0 (x , x ) and r ∈ PrfEql X 0 (x , x ). (ML 4 ) If p, q ∈ PrfEql X 0 (x, x ) and r, s ∈ PrfEql X 0 (x , x ) such that p = PrfEql X 0 (x,x ) q and r = PrfEql X 0 (x ,x ) s, then p * r = PrfEql X 0 (x,x ) q * s.
If X is a set with a proof-relevant equality, by Definition 34, we get Prf(x = X x ) := PrfEql X 0 (x, x ). Conditions (ML 1 )-(ML 3 ) express that the proof-relevant equality of X has a groupoid-structure, see Palmgren (2012a), while condition (ML 4 ) expresses the extensionality of the composition * X . Next proposition is straightforward to show.
Definition 43. Let X, Y be sets with proof-relevant equalities. A map from X to Y is a pair f := (f , f 1 ), where f : X → Y and f 1 : .
We write f : X → Y to denote a map from X to Y. We call the dependent operation f 1 the first associate of f . If, for every x, x ∈ X and every p, p ∈ PrfEql X 0 (x, x ), we have that , we say that f 1 is a function-like first associate of f . If X and Y are Martin-Löf sets, a map f : X → Y is a Martin-Löf map, if the following conditions hold: , q), for every p ∈ PrfEql X 0 (x, x ) and q ∈ PrfEql X 0 (x , x ).

Definition 44. Let I be a set with a proof-relevant equality. A family of sets over I is a triplet
:= (λ 0 , PrfEql I 0 , λ 2 ), where λ 0 : I V 0 , and λ 2 : such that the following conditions hold: (i) For every i ∈ I, we have that λ refl i ii = id λ 0 (i) .
(ii) If i = I j = I k, for every p ∈ PrfEql I 0 (i, j) and q ∈ PrfEql I 0 (j, k), the following diagram commutes (iii) If i = I j, then for for every p ∈ PrfEql I 0 (i, j), the following diagrams commute A family-map : ⇒ M is defined as in Definition 21. We denote by Fam( I) the totality of families of sets over I, which is equipped with the obvious equality. We call proof-irrelevant, if for every (i, j) ∈ D(I) and p, p ∈ PrfEql I 0 (i, j), we have that λ p ij = F(λ 0 (i),λ 0 (j)) λ p ij .
If ∈ Fam( I), then ∈ Fam * (I) (see Definition 20). If is function-like family over I, condition (iii) of the previous definition is provable, while if is proof-irrelevant, then is function-like. Following Definition 22, we denote the -set of by i∈I λ 0 (i), where and we denote the -set of , equipped with the pointwise equality, by i∈I λ 0 (i), where Proposition 45. If := (λ 0 , PrfEql I 0 , λ 2 ) is a function-like family of sets over the Martin-Löf set I, then a structure of a Martin-Löf set is defined on i∈I λ 0 (i).
If I and i∈I λ 0 (i) are Martin-Löf sets as above, it is straightforward to show that the pair pr 1 := pr 1 , 1 is a map from i∈I λ 0 (I) to I, where pr 1 : i∈I λ 0 (i) → I, (i, x) → i; i ∈ I, and 1 : , is a function-like first associate of pr 1 .
Theorem 1. Let X be a proof-relevant set, x 0 ∈ X and let PrfEql x 0 := (PrfEql x 0 0 , PrfEql x 0 1 ) be the function-like family of sets over X from Lemma 46. Let x∈X PrfEql X 0 (x, x 0 ) be equipped with its canonical structure of a Martin-Löf set, according to Proposition 45. Then for every (x, p) ∈ x∈X PrfEql X 0 (x, x 0 ), we have that (x, p) = x∈X PrfEql X Proof. By the definition of equality on the -set of some ∈ Fam( I), we have that If (x, p) ∈ x∈X PrfEql X 0 (x, x 0 ), then p ∈ PrfEql X 0 (x, x 0 ), hence x = X x 0 . If we take q := p, then PrfEql p xx 0 (p) := p −1 * p = refl x 0 .
Theorem 1 is a translation of the type-theoretic contractibility of the singleton type (see Coquand 2014) into BST. If M is the judgment (or the term) expressing this contractibility (see also Petrakis 2019d), Martin-Löf 's J-rule trivially implies M, and it is equivalent to M and the transport (see Coquand 2014). In BISH, we do not have the J-rule, but we have transport in a definitional way only. As Theorem 1 indicates, a definitional form of M is provable in BST, although there is no translation of the J-rule in BST. A map between Martin-Löf sets can generate the family of its fibers over its codomain. : is a function-like family of sets over X.

Contractible Sets and Subsingletons in BST
Next follow some results on contractible sets and subsingletons in BST that translate results from Chapters 3 and 4 of book-HoTT. According to Definition 5(iv), in BST, the truncation ||X|| of a set X is the same totality X equipped with a new equality, while in HoTT is a higher inductive type.
where the operations f * : F(Z, X) F(Z, Y) and g * : F(Z, Y) F(Z, X) are defined, respectively, by the commutativity of the following diagrams Proof. Clearly, the operations f * and g * are functions. If k ∈ F(Z, Y) and h ∈ F(Z, X), then f * (g * (k)) : Proposition 47 is an example of a result in BST the analogue of which in HoTT is shown with the axiom of univalence UA in The Univalent Foundations Program (2013) (the axiom FunExt can also be used instead).
Proposition 48. If X is a set, the following are equivalent: Proof. (i)⇒(ii) If x 0 is a centre of contraction for X, then x 0 inhabits X. If x, y ∈ X, then x = X x 0 and y = X x 0 , hence x = X y.
Remark 49. As any set can be truncated and become a subsingleton (see Definition 5(iv)), the previous proposition provided numerous examples of contractible sets. Namely, any inhabited set can be turned into a contractible set through the truncation of its equality.
(i) If : i∈I λ 0 (i) is a modulus of centres of contraction for λ 0 , i.e., i is a center of contraction for λ 0 (i), then ∈ i∈I λ 0 (i) is a center of contraction for i∈I λ 0 (i) and i∈I λ 0 (i) = V 0 I. (ii) If i 0 ∈ I is a center of contraction for I, then i∈I λ 0 (i) = V 0 λ 0 (i 0 ).
(i) If h : I i∈I λ 0 (i) is defined by h(i) := i, i , for every i ∈ I, then h is a function and (pr 1 , h) : i∈I λ 0 (i) = V 0 I. (ii) F I, i∈I λ 0 (i) = V 0 F(I, I).
(iii) If X is contractible and Y is a retract of X, then Y is contractible.
Proof. The proof of (i) is straightforward and (ii) follows from (i) and Proposition 47. For the proof of the next theorem though, we write explicitly the witnesses of the required equality in V 0 , which are the witnesses provided by the proof of Proposition 47. Let φ : F(I, i∈I λ 0 (i)) F(I, I), Clearly, φ is a function. Let θ : F(I, I) F(I, i∈I λ 0 (i)), defined by the rule g → θ(g), where θ(g) := h • g, where h is defined in case (i). Clearly, θ is a function. It is straightforward to show that (φ, θ) : F I, i∈I λ 0 (i) = V 0 F(I, I).
(iii) Let r : X → Y and s : Y → X such that r • s = id Y . It is immediate to show that if x 0 ∈ X is a center of contraction for X, then r(x 0 ) is a center of contraction for Y.
Corollary 52 is the translation in BST of the fact that UA implies the principle of weak function extensionality.
Proposition 53. Let ||X|| be the truncation of X, Y, Z subsingletons, and E a set.
Proof. (i) To show that ||λ|| ij is well defined, we use Proposition 53(iii). To show the properties of a family of sets over I for || ||, we use the corresponding properties for .

Concluding Comments
According to Feferman (see Feferman 1979, p. 207), the formal, or internal realisability interpretation of the language L(T) of a formal theory T in the language L(T ) of a formal theory T , is an assignment φ → f r φ of any formula φ of L(T) to a formula φ r :⇔ f r φ in L(T ), where φ r has at most one additional free variable f . This interpretation is sound if T φ ⇒ ∃ τ ∈Term(L(T )) T τ r φ , for every formula φ of L(T). The added axiom-scheme (A−r) "to assert is to realize" φ ⇔ ∃ f f r φ , which expresses the equivalence of the assertion of φ with its realizability, reflects the basic tenet of constructive reasoning that a statement is to be asserted only if it is proved. Note that in Feferman's refined theory with MwE, the axiom-scheme (A−r) implies the principle of dependent choice DC and the presentation axiom! (see Feferman 1979, pp. 214-215). It is also expected that one can show inductively that the scheme (A−r) is itself realisable in some theory S, i.e., In the informal, or external realisability interpretation of L(T), one defines a relation R(f , φ) between mathematical objects f of some sort and a formula φ. For example, Kleene defined such a relation for f ∈ N and φ a formula of arithmetic. External realisability interpretations can often be regarded as the reading of a formal f r φ in a specific model.
Here we described an external realizability interpretation of some part of the language of the informal theory BISH in itself, where the corresponding realisability relation is Prf(p, φ) :⇔ p ∈ Prf(φ).
Why one would choose to work within an informal framework? Maybe because to realise some formula φ does not necessarily imply that φ is constructively acceptable. For example, in Feferman (1979, pp. 207-208), Feferman defined a formal realisability interpretation of L(T 0 ) in itself such that the corresponding axiom scheme (A−r) implies the full axiom of choice. Moreover, even if one works with a realisability interpretation that avoids the realisability of the full AC, it is not certain that whatever this theory realises is constructively acceptable, or faithful to some motivating informal constructive theory like BISH. For example, the realisability of the presentation axiom in T * 0 , which holds also in the setoid-interpretation of Bishop sets in intensional MLTT, does not make it necessarily constructively acceptable. In the informal level of BISH, there is no reason to accept it.
If the main philosophical question regarding Bishop-style constructive mathematics (BCM), in general, is "what is constructive?, " an answer provided from a formal treatment of BCM that cannot be "captured" by BISH itself, is not necessarily the "right" answer.
In Feferman (1979, p. 177), Feferman criticises Bishop for a "certain casualness about mentioning the witnessing information. . . . one is looser in practice in order to keep that from getting too heavy. Practice then looks very much like everyday analysis and it is hard to see what the difference is unless one takes the official definitions seriously." In our opinion, Feferman is right on spotting this casualness in Bishop's account, which is though on purpose, as Bishop's crucial comment in Bishop (1970, p. 67) shows. One could also say that, if the difference between constructive analysis and everyday, classical analysis is difficult to see, then this is an indication of the success of Bishop's way of writing. What we find that is missing when some official definitions are not taken seriously is the proof-relevant character of Bishop's analysis and its proximity to proof-relevant mathematical analysis, like analysis within MLTT. An important consequence of revealing the witnessing information is the avoidance of choice.
The use of the axiom of choice in constructive mathematics is an indication of missing data. As we have seen already in many cases, and also in Example 9.6, the inclusion of witnessing data, like a modulus of some sort, facilitates the avoidance of choice in the corresponding constructive proof. The standard view regarding the use of choice in BISH is that some weak form of choice, countable choice, or dependent choice is necessary. This is certainly true when the witnessing data are ignored. Richman criticised the use of countable choice in BISH (see Richman 2001, andalso Schuster 2004). The revealing of witnessing data or not in BISH "oscillates" between the two extremes, regarding proof-relevance, which are also the two extremes, regarding choice. The first extreme is classical mathematics based on ZFC, where the complete lack of proof-relevance is combined with the use of a powerful choice axiom, and the second extreme is type-theoretic mathematics based on intensional MLTT, where proof-relevance is "everywhere" and the axiom of choice,i.e., the distributivity of over , is provable! When the witnessing data are ignored, then some form of weak choice is necessary for BISH, while when the witnessing data are highlighted, then choice is avoided. A similar phenomenon occurs in univalent type theory. The univalent version of the axiom of choice, in the formulation of which truncation is involved, is not provable. And what truncation does is to suppress the evidence.
Next follow some topics related to the proof-relevant character of BISH that need to be addressed in the future.
(1) A BHK-interpretation of a negated formula ¬φ is missing from Definitions 34 and 35.
As negated formulas are rare in BISH (see Petrakis 2022b; Petrakis and Wessel toappear), we find safer at the moment to exclude them from our account of a BHK-interpretation