C. Lago, We start by de ning the notion of Howe's extension, a construction extending a ?-term V-open relation to a compatible and substitutive ?-term V-relation. 2. For any a ? V if ?, x : s ? |= v a ? ? H (u, z) : ? is derivable, 2015.

, The proof is by induction on the derivation of the judgments: J ?, x : s ? |= ? a ? ? H (e, f ) : ? J ?

. H-var, x : s ? v ? (x, u) : ? ?, x : s ? |= v a ? ? H (x, u) : ? (H-var), so that s ? 1 and J is ?, x : s ? |= v a ? ? H (x, u) : ? . We have to prove a ?s ? ( v ? H ( , w )) ? ? v ? H ( , u[w/x]) : ? . Since ? is value substitutive, from ?, x : s ? |= v a ? ? H (x, u) : ? we infer a ? ? v ? (w, u[w/x]) : ? . Moreover, since ? H is an open ?-term V-relation (and thus closed under weakening), We have two subcases to consider. 1.1 J has been inferred via an instance of rule (H-var) from premises: a ? ?

H. ?-?-v-?,

, 2 J has been inferred via an instance of rule (H-var) from premises: a ? ?, : r ?

, u) : ? . We have to prove a ? s ? (? v ? H ( , w )) ? ?, : r ? v ? H ( , u[w/x]) : ? . As V is integral and ? is value-substitutive, we have: a ? s ? (? v ? H ( , w )) ? a ? ?, J is ?

, Suppose J has been inferred via an instance of rule (H-let) from premises: ?, x : s ? |= ? a ? ? H (e, e ) : ? (12.1)

?. ,

, x : s ? ) ? (?, x : r ? ) ? H (let = e in f , ) : ? . (12.3) so that is: (p ? 1) · ? ? ?, x : (p?1) ·s ?r ? |= ? (p ? 1)(a) ? b ? c ? ? H

, First of all we observe that since ? H is re exive, we can assume n = m. In fact, if e.g. n = m + l, then we can 'complete' (? H ) (m) as follows: (? H ) (m) = (? H ) (m) ·I · · · · I l -times ? (? H ) (m) · ? H · · · · ? H l -times = (? H ) (n)

, The base case is trivial. Let us turn on the inductive step. We have to prove: (s ? 1) ? e (? ? H (e, e ) : ? ) ? (? (? H ) (n) (e , e ) : ? ) ? f (?, x : s ? ? H ( f , f ) : ? ) ? (?, x : s ? (? H ) (n) ( f , f ) : ? ) ? c

, ) : ? ) ? (?, x : s ? (? H ) (n) ( f , f ) : ? ) ? c, i.e. (s ? 1) ? (? ? H (e, e ) : ? ) ? (?, x : s ? ? H ( f , f ) : ? ) ? (s ? 1) ? (? (? H

. ?-(?, x : s ? (? H ) (n) ( f , f ) : ? ) ? c

, We can now apply compatibility of ? H plus the induction hypothesis, thus reducing the thesis to: (s ? 1) · ? ? ? ? H (let x = e in f

, ? (s ? 1) · ? ? ? (? H ) T (let x = e in f , let x = e in f ) : ? ) ? c

, We can now conclude the thesis by very de nition of (? H ) T . To prove point 2 we have to show (? H ) T ? ((? H ) T ) ? . For that it is su cient to show ? H ? ((? H ) T ) ? . That amounts to prove that for all computations ? e, e : ? and values ? v , and for any a ? V such that ? |= ? a ? ? H (e, e ) : ? is derivable we have a ? ? (? H ) T (e, e ) : ? (and similarity for ? v , : ? ). The proof is by induction on the derivation of ? |= ? a ? ? H

, If any CBE in ? is nitely continuous, Theorem 17

. T-=-?, One inequality follows from Lemma 41 as follows: ? ? ? H ? (? ) T . For the other inequality, we rely on the coinduction proof principle associated with ? , meaning that it is su cient to prove that ((? ) H ) T is a symmetric applicative simulation (with respect to ?). Symmetry is given by Lemma 45. From Lemma 43 (Key Lemma) we know that ? H is an applicative simulation distance. Since the identity ?-term V-relation is an applicative simulation distance, and the composition of applicative simulation distances is itself an applicative simulation distance (see Proposition 30), Lemma 45 we know that (? H ) T is compatible. Therefore, it is su cient to prove

, We conclude this chapter by noticing that we can rely on Theorem 17 to come up with concrete notions of compatible applicative bisimilarity distance. For instance, we obtain a compatible pseudometric for for P-Fuzz (observe that CBEs are indeed nitely continuous)

. Koutavas, normal form bisimilarity has been mainly investigated for call-by-name languages, due to its equivalence with Lévy-Longo tree equivalence, and for the so-called classical theory of ?-calculus (Barendregt, 1984), due to its equivalence with Böhm tree equivalence. To the best of the author's knowledge, no abstract account of normal form bisimilarity in the context of languages with algebraic e ects has been given so far. However, there are some works on speci c e ectful extensions of Böhm trees, notably the work of de Liguoro and Piperno on nondeterministic Böhm trees, de ned are proved using our abstract Howe's method. Applicative bisimilarity has also been investigated for languages with non-algebraic e ects, notably for languages with control operators (Biernacki & Lenglet, 1993.

, Similar in spirit to our approach, the work of Katsumata and Sato (Katsumata & Sato, 2013) (as well as (Katsumata, 2013)) analyses monadic lifting of relations in the context of -lifting

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