Realizability for Peano Arithmetic with Winning Conditions in HON Games

We build a realizability model for Peano arithmetic based on winning conditions for HON games. First we define a notion of winning strategies on arenas equipped with winning conditions. We prove that the interpretation of a classical proof of a formula is a winning strategy on the arena with winning condition corresponding to the formula. Finally we apply this to Peano arithmetic with relativized quantifications and give the example of witness extraction for \(\Pi^0_2\)-formulas.


Introduction
Realizability is a technique to extract computational content from formal proofs. It has been widely used to analyze intuitionistic systems (for e.g. higher-order arithmetic or set theory), see [1] for a survey. Following Griffin's computational interpretation of Peirce's law [2], Krivine developed in [3][4][5] a realizability for second-order classical arithmetic and Zermelo-Fraenkel set theory.
On the other hand, Hyland-Ong game semantics provide precise models of various programming languages such as PCF [6] (a similar model has simultaneously been obtained in [7]), also augmented with control operators [8] and higher-order references [9]. In these games, plays are interactions traces between a program (player P) and an environment (opponent O). A program is interpreted by a strategy for P which represents the interactions it can have with any environment.
In this paper, we devise a notion of realizability for HON general games based on winning conditions on plays. We show that our model is sound for classical Peano arithmetic and allows to perform extraction for Π 0 2 -formulas. HON games with winning conditions on plays have been used in e.g. [10] for intuitionistic propositional logic with fixpoints. Our winning conditions can be seen as a generalization of the ones of [10] in order to handle full first-order classical logic, while [10] only deals with totality. Our witness extraction is based on a version of Friedman's trick inspired from Krivine [4]. Classical logic is handled similarly to the unbracketed game model of PCF of [8].
We start from the cartesian closed category of single-threaded strategies which contains the unbracketed and non-innocent strategies used to model control operators and references. We use a category of continuations in the coproduct completion of [11], so that the usual flat arena of natural numbers in HON games is indeed in the image of a negative translation. Our realizability is then obtained by equipping arenas with winning conditions on plays.
The paper is organized as follows. Section 2 recalls the game semantics framework and how to interpret λµ-calculus in it. Section 3 defines the notion of winning strategies. Section 4 contains the definition of our realizability relation and its adequacy for classical logic. Section 5 applies our realizability model to Peano arithmetic and shows witness extraction for Π 0 2 -formulas.

HON Games
Our realizability model is based on the Hyland-Ong-Nickau games [6] with no bracketing or innocence constraint, so as to model control operators and references [8,9]. We consider single-threaded strategies in order to have a cartesian closed category.

Arenas and Strategies Definition 1 (Arena). An arena is a countable forest of moves. Each move is given a polarity O (for Opponent) or P (for Player or Proponent):
-A root is of polarity O.
-A move which is not a root has the inverse polarity of that of his parent.
A root of an arena is also called an initial move. We will often identify an arena with its set of moves.
Definition 2 (Justified sequence). Given an arena A, we define a justified sequence on A to be a word s (finite or infinite) of A together with a partial justifying function f : |s| ⇀ |s| such that: We denote the empty justified sequence by ǫ. Remark here that by definition of the polarity, if f (i) is undefined (s i is initial), then s i is of polarity O, and if f (i) is defined, then s i and s f (i) are of opposite polarities. Also, f (0) is never defined, and so s 0 is always an initial O-move. A justified sequence is represented for example as: A subsequence of a justified sequence s is a subword of s together with a justifying function defined accordingly. In particular if a move a points to a move b in the original sequence and if a is in the subsequence but b is not, then the pointer from a is left undefined. For example the following sequence is a subsequence of the one above: If A is an arena, X is a subset of A and s is a justified sequence on A, then s |X is the subsequence of s consisting of the moves of s which are in X.
In a sequence s, a move s j is hereditarily justified by a move s i if s i is initial and for some n, f n (j) = i. Warning. Note that a thread is a justified sequence which may not be alternating, so our definition of thread differs from the usual one. By extension a justified sequence s will be called a thread if it contains exactly one thread (i.e. Threads(s) = {s}). Remark that Threads(ǫ) = ∅ and so ǫ is not a thread. A P -sequence (resp. O-sequence) is a sequence ending with a P -move (resp. a O-move). Write t ⊑ s if t is a prefix of s, i.e. t is a prefix of s as a word and their justifying functions coincide (this is a particular case of subsequence). Write t ⊑ P s (resp. t ⊑ O s) if t is a P -prefix (resp. O-prefix) of s, i.e. t ⊑ s and t is a P -sequence (resp. O-sequence).

Definition 4 (Play).
A play s on A is an alternating justified sequence of A, i.e., for any i, s 2i is a O-move and s 2i+1 is a P -move. We denote the set of plays of A by P A .
A play on an arena is the trace of an interaction between a program and a context, each one performing an action alternatively. A P -play (resp. O-play) is a play which is a P -sequence (resp. O-sequence).

Definition 5 (Strategy).
A strategy σ on A is a P -prefix-closed set of finite P -plays on A such that: σ is deterministic: if sm and sm ′ are in σ, then m = m ′ .
Our notion of single-threadedness matches the usual one of thread-independence (see e.g. [9]). Remark also that a strategy always contains the empty play ǫ since T hreads(ǫ) = ∅.

Cartesian Closed Structure
The constructions we use will sometimes contain multiple copies of the same arena (for example A → A), so we distinguish the instances with superscripts (for example A (1) → A (2) ).
Let U be the empty arena and V be the arena with only one (opponent) move. If A and B are arenas consisting of the trees A 1 . . . A p and B 1 . . . B q , then the arenas A → B and A × B can be represented as follows: A → B : A × B : The constructions described here define a cartesian closed category whose objects are arenas and morphisms are strategies. Details of the construction can be found in [12]. In the following this category will be denoted as C. These definitions of arenas will be used to associate arenas to the following simple types: Definition 6 (Simple types). The simple types are defined by the following grammar, where ι ranges over a set of base types: Since C is cartesian closed, we use the syntax of λ-calculus to define strategies from other strategies. In order to distinguish this notation from the λµ-terms of Sect. 2.3 we use a bold lambda λ.

Interpretation of the call-by-name λµ-calculus
We map classical proofs to strategies using the interpretation of call-by-name λµ-calculus in categories of continuations described in [13]. In order to make explicit the double negation translation of the base types, we base the model on the category of continuations R Fam(C) , where the response category Fam(C) is a variant of the coproduct completion described in [11] applied to the category C defined in Sect. 2.2: Definition 7 (Fam(C)). The objects of Fam(C) are families of objects of C indexed by at most countable sets, and a morphism from Remark here that we differ from [11] because C doesn't have weak coproducts nor all small products, and the families are countable. Thus Fam(C) is not bicartesian closed, but since C is cartesian closed and has countable products, Fam(C) is still a distributive category with finite products and coproducts, and has exponentials of all singleton families. The empty product and terminal object is the singleton family {1}, the empty sum and initial object is the empty family {}, and: We fix once and for all: which is an object of Fam(C) as a singleton family. R has all exponentials as stated above. Note that the canonical morphism δ A : The category of continuations R Fam(C) is the full subcategory of Fam(C) consisting of the objects of the form R A . The objects of R Fam(C) are singleton families, and R Fam(C) is isomorphic to C. We will consider that objects and morphisms of R Fam(C) are arenas and strategies and we will use the vocabulary defined at the end of Sect. 2.2 on R Fam(C) also.
Interpreting the call-by-name λµ-calculus. The types of λµ-calculus are the simple types of Definition 6. Let k T range over a set of typed constants and x T (resp. α T ) range over a countable set of variables (resp. names) for each type T . The grammar of λµ-terms is the following: The typing rules can be found in [13], where our unit is their ⊤, our × is their ∧ and our void is their ⊥. For instance, the Law of Peirce is the type of the following term (we omit the type annotation of the variables).
This λµ-term will be denoted cc.
We follow [13] to interpret call-by-name λµ-calculus in R Fam(C) . In partic- ]. This morphism coincides with the interpretation of the call-by-name CPS translation of M (defined in [13]) in the cartesian closed category R Fam(C) . See [13] for details. As stated in [13], if the call-by-name CPS translations of two terms are βη-equivalent, then their interpretations are the same.
In the following we will drop the double brackets for the interpretation of simple types.

Winning Conditions on Arenas
We will now define our notion of realizability. We equip arenas with winning conditions on threads. Realizers are then winning strategies, intuitively strategies which threads are all winning.
It is well-known that preservation of totality by composition of strategies is problematic in game semantics. Luckily we do not need to preserve totality, but only winningness. We thus do not impose any totality condition on strategies, but when it turns to the definition of winning threads, we have to take into account all maximal threads, including both infinite and odd-length threads. This leads to the notion of winning strategy proposed in Definition 12.
In order to define the notion of winning condition on an arena we introduce the notion of P -subthread and O-subthread: If t is a thread and u is a subsequence of t which is a thread, then u is a: Now we can define the notion of winning condition on an arena:

Definition 9 (Winning condition). A winning condition on A is a set W of threads on A such that:
-If t is a thread on A and if some P -subthread of t is in W, then t ∈ W.
-If t ∈ W then all the O-subthreads of t are in W.
A justified sequence s on the arena A equipped with the winning condition W is said to be winning if Threads(s) ⊆ W.
Our notion of winning sequence can be seen as a generalization of the one defined in [10]. In order to obtain a realizability model of first-order logic, the notion of winning sequence is non-trivial and there can be odd-length sequences which are winning and even-length sequences which are losing. Remark that if t is a thread on A → B, then t |B is a thread on B, so t |B is winning iff t |B ∈ W B , and if t is a thread on A × B, then t is either a thread on A, either a thread on B.
Definition 10 (Arrow and product of winning conditions). If W A and W B are sets of threads on the arenas A and B, then we define: Winning Strategies. In order to define what is a winning strategy, we use a notion of augmented plays of a strategy inspired from [14]: Definition 11 (Augmented play). If σ is a strategy on A, then s is an augmented play of σ if one of the following holds: s ∈ σ, or s ∈ P A is such that ∀t ⊑ P s, t ∈ σ and ∀t ∈ σ, s ⊑ t.
In particular, in the second case of the above definition, s is either a O-sequence, either an infinite sequence (in which case s ⊑ t ⇔ s = t and so the second condition, equivalent to s / ∈ σ, is always true). Remark that unlike [14], we consider not only odd-length extensions (with an O-move), but also infinite ones.
Definition 12 (Winning strategy). If σ is a strategy on the arena A equipped with the winning condition W, then σ is said to be winning if all its augmented plays are winning.
The following lemma will be useful to prove that a strategy σ is winning on (A, W).

Lemma 2. If σ is a strategy on A and if s is an augmented play of σ, then every t ∈ T hreads(s) is an augmented play of σ.
Using this lemma it is sufficient to prove that every augmented play of σ which is a thread (let us call it an augmented thread of σ) is in W A in order to prove that σ is winning on (A, W A ).
We now prove that the winning conditions on the arrow and product are compatible with application and pairing of strategies. Proof. Let t be an augmented thread of σ(τ ). By definition of composition of strategies, there is some augmented play u of σ such that u |A is an augmented play of τ and u |B = t. Since t is a thread, u is also a thread, so since σ is winning on A → B, u ∈ W A→B . u |A is an augmented play of τ which is winning on A, so u |B is winning, and so t = u |B is a winning thread: t ∈ W B . Therefore σ(τ ) is winning.
⊓ ⊔ Proof. Let t be an augmented thread of σ(τ ). By definition of product of strategies, t |A is an augmented play of σ and t |B is an augmented play of τ , so since σ and τ are winning, t |A and t |B are winning, and so t ∈ W A×B . Therefore σ, τ is winning.

⊓ ⊔
The following technical lemma on the interpretation of cc will be useful.

Lemma 5. If t is an augmented thread of [[cc]] on the arena
, then the threads of t |T (1) and t |T (2) are P -subthreads of t |T (3) .
It follows easily from this lemma and Lemma 1 that for any winning conditions W T and W U , [[cc]] is winning on the arena Remark on the arrow on winning conditions. Let A, B be arenas equipped with winning conditions W A , W B . We define here a strategy σ on A → B such that for any winning strategy τ on A, σ(τ ) is winning on B, but σ is not winning on A → B. Hence the arrow on winning conditions differs from the usual Kleene realizability arrow (see [1]). We choose A and B to be the same arena Q consisting of one root with three children ♯, ♭ and ♮, equipped with the winning condition where the threads may be finite or infinite. We define a strategy σ on Q → Q such that for any τ winning on (Q, W Q ), σ(τ ) is winning on (Q, W Q ), but σ is not winning on (Q → Q, W Q→Q ). σ is the innocent strategy defined by the views: where a and b are distinct moves. The interaction with any single threaded strategy will produce the left view, and so the projection q O ♯ P will be winning, but the right view (which will never happen in an interaction with a singlethreaded strategy) with a = ♯ and b = ♮ is losing, so σ is losing.

First-order Logic
We define a realizability model for first-order classical logic with possibilities of witness extraction. For that the proposition ⊥ will be mapped to an arena ι in general different from V. Its associated winning condition will be a parameter of the model, in the spirit of [4]. Let x range over a countable set of variables, f range over a set of function symbols with fixed finite arity and P range over a set of predicate symbols with fixed finite arity. First-order terms and formulas are defined by the following grammar: a, b := x | f (a 1 , . . . , a n ) In the following we use syntactic sugar for the negation of formulas: ¬A ≡ A ⇒ ⊥ and for the existential: ∃xA ≡ ¬∀x¬A. We fix a countable first-order structure interpreting the terms of our logic, that is a countable set E together with an interpretation f E : E n → E for each function symbol. The interpretation is extended to every closed term: if a is a closed term of the logic, then a E denotes its interpretation in the first-order structure, so a E is an element of E.

Realizability
We let ⊥ ⊥ be an arbitrary subset of E. We consider simple types with a type constant P * for each predicate P and a type constant ι to interpret ⊥. We can map any first-order formula A to such a simple type A * as follows: (P (a 1 , . . . , a n )) * = P * ⊤ * = unit Remark that the type ⊥ * is not the type void because the associated arena would be too small to hold informational content. Recall that we omit the double bracket notation for the arenas, so a type T also denotes the associated arena. We suppose that for each atomic predicate P , the type P * comes with its associated arena. We fix the arena associated to ι to be R (R E ) , where E = {U e | e ∈ E} is the countable family of empty arenas (and R = {V}). Hence ι is the usual flat arena for the set E.
Let us suppose we associate to each predicate P (a 1 , . . . , a n ) with a 1 , . . . , a n closed first-order terms a winning condition W P (a1,...,an) on the arena P * . We can then define for each closed first-order formula A a winning condition W A on the arena A * . The winning conditions W A∧B and W A⇒B are as in Definition 10, and we let: Note that these are indeed winning conditions. For W ⊤ , the empty set is a winning condition on U which is the empty arena with no thread. For W ∀xA , it is easy to see that an intersection of winning conditions is a winning condition. For W ⊥ , the thread q O m P 1 m P 2 . . . (that may be finite or infinite) has only itself as O-subthread and q O m P i1 m P i2 . . . for 1 ≤ i 1 < i 2 ≤ . . . as P -subthreads so W ⊥ is a winning condition on ι.
We can now define our notion of realizability:

Definition 13 (Realizability relation). If A is a closed first-order formula and if σ is a strategy on
The following lemma shows that the identity formulas are realized by the corresponding identity strategies.

Lemma 6. If A is a closed formula, then the identity strategy id A * on A * is a realizer for the formula
is also winning, and so t ∈ W A * .

⊓ ⊔
The following result is a consequence of the remark following Lemma 5.

Adequacy for Minimal Classical Logic
We now show that realizability is compatible with deduction in minimal classical logic. Full classical logic is discussed in Sect. 4.3.
Deduction system. Let Ax be a set of closed formulas. We use the following deduction system based on natural deduction with a rule for the law of Peirce, where Γ is a sequence of formulas A 1 , . . . , A n .
Remark that ⊥ has no associated rule, since the ex-falso rule has a particular status, given the interpretation of ⊥. This will be discussed in Sect. 4.3.
Translation of proofs to strategies. We use λµ-calculus and its interpretation in R Fam(C) to map a first-order proof to a typed λµ-term which is then interpreted in R Fam(C) as a strategy.
Assume given a constant k A of type A * for each A ∈ Ax. We map a derivation ν of A 1 , . . . , A n ⊢ A to a typed λµ-term ν * of type A with free variables among x A * 1 , . . . , x A * n as follows: Adequacy. We now prove that the strategies interpreting the proofs are realizers of the proved formula. If A is a formula and θ an assignment of terms to variables, then θ(A) denotes A where all the free variables are replaced with their image by θ. Proof (sketch). The case of the variable follows from Lemma 6. That of cc comes from Lemma 5. Product introduction is dealt with using Lemma 4, and arrow elimination using Lemma 3. The other cases are straightforward. ⊓ ⊔

Full Classical Logic
In order to get full classical logic we need to add an ex-falso rule. However since the arena ⊥ * is not empty (see Sect. 4.1), we restrict ex-falso to a certain class of formulas. We have to ensure that (ι, W ⊥ ) is included in (A * , W A ). This means that ι is a subtree of A * , so a play on ι is in particular a play on A * , and that W ⊥ ⊆ W A . We will call these formulas explodable since they satisfy the principle of explosion. We add to our deduction system the following rule: In particular any formula ending with ⊥ is explodable, where a formula ending with ⊥ is a formula generated by the grammar: where A is any first-order formula (defined in Sect. 4). The corresponding adequacy lemma is immediate from Lemma 8.

First-order Logic with Equality
We now show how to handle equality. We suppose that our first-order language contains an inequality predicate = of arity 2 interpreted by the simple type ι (see Sect. 4.1). The associated winning condition is: (recall that E is the first-order structure chosen at the beginning of Sect. 4). It is easy then to verify that any formula ending with the predicate a = b is explodable. In the following we use the notation (a = b) ≡ ¬(a = b). The axioms for equality are: Recall that ∀x(x = x) is only syntactic sugar for ∀x(x = x ⇒ ⊥), and that ∀x∀y( Lemma 9. Let ⊥ ⊥ ⊆ E. 1. The identity strategy on ι, is a realizer of (ref l).

Peano Arithmetic
We now proceed to the realizability interpretation of full Peano arithmetic.

Definitions
Our first-order language is built from the function symbols 0 of arity 0, S of arity 1 and + and × of arity 2. The predicate symbols are = of arity 2 and nat of arity 1. This choice of function symbols is only for simplicity, and we could choose to have all the symbols of primitive recursive functions. We also fix the structure interpreting the terms of the logic to be the set of natural numbers N. The symbols 0, S, + and × are interpreted the standard way. The typed λµ-calculus in which we interpret the proofs has ι as unique base type. All the predicate symbols and ⊥ are interpreted as ι, and the associated arena in R Fam(C) is [[ι]] = R (R N ) where N = {U n | n ∈ N} (see Sect. 4.1). Hence the type of natural numbers is interpreted as the negative translation of N. Note that this is the usual flat arena of natural numbers: This differs from Laird's interpretation of PCF with control [15], where the base type of natural numbers is interpreted by the arena (ι → ι) → ι. The winning conditions for ⊥ and a = b are as in Sects. 4.1 and 4.4, and the winning condition for nat(a) is: which is a winning condition, using the same arguments as for W ⊥ . From this we can check that every formula which contains no nat(a) predicate at rightmost position is explodable. We use the following syntactic sugar: The relativization A n of a formula is defined as the identity on all constructions except for the quantification: (∀xA) n ≡ ∀ n xA n . Note that if a formula does not contain any nat(a) predicate, then its relativization has no nat(a) predicate at rightmost position, so it is explodable. The axioms are the ones for equality (defined in Sect. 4.4) and the universal closures of: We will now define the realizers for these axioms. We first define the strategies computing basic operations and recursion on natural numbers.
In Fam(C) a morphism from ⊤ * = {U} to N = {U n | n ∈ N} is given by a function from the singleton set to N together with a strategy from U to U . Since there is only one such strategy, such a morphism is given by a natural number. We will call this morphism τ n . Similarly a morphism from N k to N is given by a function from N k to N. This leads to morphisms τ S , τ + and τ × respectively on N → N, N → N → N and N → N → N. From these we define the following morphisms of R Fam(C) : σ n = λk.kτ n : ι σ S = λn.λk.n(λn ′ .k(τ S n ′ )) : ι → ι σ + = λmλn.λk.m(λm ′ .n(λn ′ .k(τ + m ′ n ′ ))) : ι → ι → ι σ × = λmλn.λk.m(λm ′ .n(λn ′ .k(τ × m ′ n ′ ))) : ι → ι → ι The above morphisms correspond to the expected strategies: Lemma 10. The strategies σ n , σ S , σ + and σ × are the innocent strategies defined by the views: We now move to the definition of ρ T , the recursor on type T , which is the usual recursor of Gödel's system T. For that we define for each n ∈ N and simple type T a strategy ρ T n by: and we finally define the strategy ρ T as the innocent strategy which views are: We use the following lemma in order to prove the validity of (ind): Proof. This is an immediate consequence of Lemma 8, since the strategies ρ T 0 and ξ T are the interpretations of proofs of the formulas.

ρ A * is a realizer of (ind)
Proof (sketch). The cases 1 and 2 are straightforward. We prove cases 3, 4, 5 and 6 using Lemma 10. For 7 we prove by induction on n that: using Lemma 11. We finally prove that ρ A * is a realizer of the (ind) axiom for formula A. Let t be an augmented thread of ρ A * on the arena . We want to prove that t |ι (2) →A (4) is winning on ∀x n A[x], so let a be a closed first-order term, let n = a N and let suppose that t |ι (2) is winning on nat(a). Then there must be some n O in t |ι (2) . Let u be the subsequence of t consisting of the initial q O , the following q P , this n O and all the moves m of t such that the view obtained immediately after m contains n O . Then u is a play of ρ A * n . Since a P -move does not change the current view, the threads of u |A (1) ⊓ ⊔ Theorem 1. If A is provable in Peano arithmetic then there is a computable strategy σ such that σ A n .

Extraction
We now show that from any provable Π 0 2 -formula we can extract a computable witnessing function.
Suppose that we have a proof of ⊢ ∀ n x∃ n y(a = b). We obtain by doublenegation elimination a proof of ⊢ ∀ n x(¬∀ n y(a = b)), and we map it to a strategy σ such that: and therefore σ(σ n )(id ι ) ⊥. Then we can prove that σ(σ n )(id ι ) is some σ m such that m ∈ ⊥ ⊥. Indeed, if σ(σ n )(id ι ) is the empty strategy then its only augmented play is q O , which is losing on ⊥.

Conclusion & Future Work
We have built a realizability model for Peano arithmetic using winning conditions on arenas, and have used it in the context of witness extraction for Π 0 2 -formulas. Future work will be the comparison of the present model with the game interpretation of classical arithmetic of [16], and with the winning conditions on sequential games of [17] and [18]. Our main goal is to compare two different versions of realizers for the axiom of dependent choices: the modified bar recursion of [19] and the clock of [3].
Lemma 13. If W A and W B are winning conditions on A and B, then W A→B is a winning condition on A → B and W A×B is a winning condition on A × B.
Proof. -Let t be a thread on A → B and let u be a P -subthread of t such that Then since u is winning, u |B is winning and is a P -subthread of t |B which is therefore also winning. Finally t is winning. -Let t be a thread on A → B such that t ∈ W A→B and let u be a O-subthread of t. Let suppose that T hreads(u |A ) ⊆ W A . If v ∈ T hreads(t|A) and if m is the initial move of v, then m points to the initial move of t which is a O-move, and since u is a O-subthread of t, we get m ∈ u. Now the thread of u |A which initial move is m is in W A and is a P -subthread of v, so v ∈ W A . Therefore T hreads(t|A) ⊆ W A , and since t ∈ W A→B we have t |B ∈ W B , and since u |B is a O-subthread of t |B , we get u |B ∈ W B . Finally u ∈ W A→B . -Let t be a thread on A × B. t is either a thread on A, either a thread on B, so if u is a winning P -subthread of t, then either u ∈ W A , either u ∈ W B . Therefore t ∈ W A or t ∈ W B , and so t ∈ W A×B . -Let t be a thread on A × B such that t ∈ W A×B . Either t ∈ W A , either t ∈ W B , so any O-subthread of t is in W A or W B , so in W A×B . ⊓ ⊔ Lemma 14. If σ is a strategy on A and if s is an augmented play of σ, then every t ∈ T hreads(s) is an augmented play of σ. Proof.
-If s is an O-sequence, then we write s = s ′ m with s ′ ∈ σ. Let t ∈ T hreads(s).
If m is not a move in t, then t ∈ T hreads(s ′ ) ⊆ σ. If m is a move in t, then we write t = t ′ m, so t ′ ∈ T hreads(s ′ ) ⊆ σ. If there is some n such that tn = t ′ mn ∈ σ, then T hreads(s ′ mn) = (T hreads(s ′ ) \ {t ′ }) ∪ {t ′ mn} ⊆ σ, so by single-threadedness of σ, sn = s ′ mn ∈ σ, contradicting the fact that s is an augmented play of σ. -If s is infinite, let t ∈ T hreads(s). If t is finite, then there is some s ′ ⊑ P s such that t ∈ T hreads(s ′ ), but s ′ ∈ σ, so by single-threadedness of σ t ∈ σ. If t is infinite, then for all t ′ ⊑ P t there is some s ′ ⊑ P s such that t ′ ∈ T hreads(s ′ ), but s ′ ∈ σ, so by single-threadedness of σ t ∈ σ. ⊓ ⊔ Fam(C) satisfies the mono requirement and therefore Fam(C) is a response category(see [13]): Lemma 15 (mono requirement). If C is the category defined in Sect. 2.2, then the canonical morphism ∂ A : A → R (R A ) in Fam(C) is monic.
Proof. Let ϕ : B → A. If we write A = {A i | i ∈ I} and B = {B j | j ∈ J} in Fam(C), then ϕ consists of a function f ϕ : J → I together with a family of strategies ϕ j : B j → A fϕ(j) .
∂ A consists of a family of strategies ∂ Ai0 on A i0 → (Π i∈I (A i → R)) → R which plays are copycat plays on the subarena A i0 → (A i0 → R) → R. Therefore, the plays of the jth fiber of ϕ; ∂ A are in the subarena B j → (A fϕ(j) → R) → R and their projections on B j → A fϕ(j) are exactly the plays of ϕ j . In other words: ϕ j = {s |Bj →A fϕ (j) | s ∈ (ϕ; ∂ A ) j } therefore, if ϕ; ∂ A = ψ; ∂ A , then ϕ = ψ, so ∂ A is monic. Proof. We prove the property by induction on the proof tree: