By Sam Lindley, Conor McBride, Phil Trinder, Don Sannella
This quantity is released in Honor of Philip Wadler at the social gathering of his sixtieth birthday, and the gathering of papers shape a Festschrift for him. The contributions are made by way of a number of the many that be aware of Phil and feature been encouraged by way of him. The examine papers incorporated the following symbolize a few of the components within which Phil has been lively, and the editors thank their colleagues for agreeing to give a contribution to this Festschrift. we strive to summarize Phil Wadler's clinical achievements. moreover, we describe the non-public variety and exuberance that Phil has dropped at the subject.
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Extra info for A List of Successes That Can Change the World: Essays Dedicated to Philip Wadler on the Occasion of His 60th Birthday
S1 B , sp 2 :: A S2 B , if sp 1 ≡b sp 2 ♦ Proof. right. This result is surprising because the two equivalences come from rather diﬀerent perspectives. Johnson and Rosebrugh introduced a form of span equivalence, and showed that it implies bisimulation equivalence. They did not explicitly address the question of whether this implication is strict. However, there Reﬂections on Monadic Lenses 17 are some diﬀerences between their presentation and ours; the most important diﬀerence is the fact that we assume lenses to be equipped with a create function, while they consider lenses without create functions but sometimes consider spans of lenses to be “pointed”, or equipped with designated initial state values.
A bisimulation of M -lens spans sp 1 :: [A S1 B ]M and sp 2 :: [A S2 B ]M is a M -lens span B ]M where R ⊆ S1 × S2 and fst is a base map from sp to sp 1 sp :: [A R and snd is a base map from sp to sp 2 . We write sp 1 ≡b sp 2 when there is a ♦ bisimulation of spans sp 1 and sp 2 . Figure 2 illustrates the three equivalences diagrammatically. 7. Each of the relations ≡i , ≡s and ≡b are equivalence relations on compatible spans of M -lenses and satisfy (Identity), (Assoc) and (Cong). 8. sp1 ≡i sp2 implies sp1 ≡s sp2 , but not the converse.
Instead, we construct the appropriate span as follows. mget (s2 )}. In the absence of dependent types, we represent this type as (S1 , S2 ) in Haskell, and we need to check that the mput and mcreate operations respect the consistency invariant. 8. If ml 1 :: [S1 B ]M and ml 2 :: [S2 S2 ]μ . so is ml 1 ml 2 :: [S1 (S1 S2 ) B ]M are well-behaved then ♦ Note that (MPutGet) and (MCreateGet) hold by construction and do not need the corresponding properties for l1 and l2 , but these properties are needed to show that consistency is established by mcreate and preserved by mput.