minor correction

sofp-src/sofp-applicative.lyx

@@ -276,18 +276,10 @@ Motivation for applicative functors
 \end_layout
 
 \begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-vspace{-0.25cm}
+We would like to parallelize independent computations
 \end_layout
 
-\end_inset
-
+\begin_layout Standard
 Monads are inconvenient for expressing 
 \emph on
 independent
@@ -452,10 +444,6 @@ Future { c1 }.flatMap { x
 
 \end_layout
 
-\begin_layout Standard
-We would like to parallelize independent computations
-\end_layout
-
 \begin_layout Standard
 We would like to accumulate 
 \emph on
@@ -634,7 +622,10 @@ for {
 \end_layout
 
 \begin_layout Standard
-We would like to express a computation where effects are unordered
+We would like to express a computation where effects are unordered and /
+ or do not depend on previous results.
+ flatMap cannot express independent effects even if we use flatMap(_ =>
+ ...)
 \end_layout
 
 \begin_layout Standard
@@ -721,18 +712,6 @@ map3
 \end_layout
 
 \begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-vspace{-0.15cm}
-\end_layout
-
-\end_inset
-
 Consider 1, 2, 3, ...
  commutative and independent 
 \begin_inset Quotes eld
@@ -1104,6 +1083,10 @@ status open
 \end_inset
 
 
+\family default
+\size default
+\color inherit
+
 \begin_inset ERT
 status open
 
@@ -1116,6 +1099,13 @@ hrule
 
 \end_inset
 
+Note that the map2 function defined via flatMap will have the right type
+ signature and will work.
+ But sometimes we need a different implementation, not equivalent to an
+ implementation via flatMap.
+\family typewriter
+\size footnotesize
+\color blue
 
 \begin_inset VSpace defskip
 \end_inset
@@ -5332,7 +5322,7 @@ contramap
 Examples of profunctors: 
 \size footnotesize
 
-\begin_inset Formula $P^{A}\triangleq\bbnum1+\text{Int}\times A\rightarrow A$
+\begin_inset Formula $P^{A}\triangleq\bbnum 1+\text{Int}\times A\rightarrow A$
 \end_inset
 
 ; 
@@ -6231,7 +6221,7 @@ pure
 
 \begin_layout Enumerate
 Show that 
-\begin_inset Formula $F^{A}\triangleq\left(A\rightarrow Z\right)\rightarrow\bbnum 1+A$
+\begin_inset Formula $F^{A}\triangleq\left(A\rightarrow Z\right)\rightarrow\bbnum1+A$
 \end_inset
 
  is a functor but not applicative.
@@ -6255,7 +6245,7 @@ Show that
 
 \begin_layout Enumerate
 Implement an applicative instance for 
-\begin_inset Formula $F^{A}=\bbnum 1+\text{Int}\times A+A\times A\times A$
+\begin_inset Formula $F^{A}=\bbnum1+\text{Int}\times A+A\times A\times A$
 \end_inset
 
 .
@@ -6297,7 +6287,7 @@ For any contrafunctor
 
 \begin_layout Enumerate
 Show that the recursive functor 
-\begin_inset Formula $F^{A}\triangleq\bbnum 1+G^{A\times F^{A}}$
+\begin_inset Formula $F^{A}\triangleq\bbnum1+G^{A\times F^{A}}$
 \end_inset
 
  is applicative if