winitzki · winitzki · Oct 17, 2025 · Nov 29, 2024 · Nov 29, 2024 · Nov 30, 2024
diff --git a/sofp-src/lyx/sofp-appendices.lyx b/sofp-src/lyx/sofp-appendices.lyx
diff --git a/sofp-src/lyx/sofp-applicative.lyx b/sofp-src/lyx/sofp-applicative.lyx
diff --git a/sofp-src/lyx/sofp-curry-howard.lyx b/sofp-src/lyx/sofp-curry-howard.lyx
diff --git a/sofp-src/lyx/sofp-disjunctions.lyx b/sofp-src/lyx/sofp-disjunctions.lyx
diff --git a/sofp-src/lyx/sofp-filterable.lyx b/sofp-src/lyx/sofp-filterable.lyx
diff --git a/sofp-src/lyx/sofp-free-type.lyx b/sofp-src/lyx/sofp-free-type.lyx
diff --git a/sofp-src/lyx/sofp-functors.lyx b/sofp-src/lyx/sofp-functors.lyx
diff --git a/sofp-src/lyx/sofp-higher-order-functions.lyx b/sofp-src/lyx/sofp-higher-order-functions.lyx
diff --git a/sofp-src/lyx/sofp-induction.lyx b/sofp-src/lyx/sofp-induction.lyx
@@ -19494,9 +19494,8 @@ What problems cannot be solved with these tools?
 \begin_layout Standard
 There is no automatic recipe for converting an arbitrary function into a
  tail-recursive one.
- The accumulator trick does not always work! In some cases, it is impossible
- to implement tail recursion in a given recursive computation.
- An example of such a computation is the 
+ The accumulator trick does not always work! An example where it is impossible
+ to implement tail recursion is the 
 \begin_inset Quotes eld
 \end_inset
 
@@ -20365,7 +20364,20 @@ def countWords(s: String): Map[String, Int] = {
 
 \end_inset
 
-An alternative, shorter implementation of the same function is:
+An alternative, shorter implementation of the same function can be written
+ using the method 
+\begin_inset listings
+inline true
+status open
+
+\begin_layout Plain Layout
+
+groupBy
+\end_layout
+
+\end_inset
+
+ like this:
 \begin_inset listings
 inline false
 status open
@@ -20390,7 +20402,7 @@ def countWords(s: String): Map[String, Int] =
 
 \end_inset
 
-The 
+This code creates a dictionary by a single function call to 
 \begin_inset listings
 inline true
 status open
@@ -20402,7 +20414,7 @@ groupBy
 
 \end_inset
 
- creates a dictionary in one function call rather than one entry at a time.
+.
  But the resulting dictionary contains word lists instead of word counts,
  so we use 
 \begin_inset listings
@@ -21314,10 +21326,11 @@ n: Int
 \end_inset
 
  for 
-\begin_inset Formula $k>0$
+\begin_inset Formula $k=1$
 \end_inset
 
-, where 
+, 2, ...
+ Here 
 \begin_inset Formula $SD(x)$
 \end_inset
 
@@ -23967,7 +23980,7 @@ status open
 
 \begin_layout Plain Layout
 
-def revSentence(s: String): String = ???
+def revSentence(text: String): String = ???
 \end_layout
 
 \begin_layout Plain Layout
@@ -24151,38 +24164,14 @@ noprefix "false"
 \end_layout
 
 \begin_layout Standard
-Define a function 
+Denote 
 \begin_inset listings
 inline true
 status open
 
 \begin_layout Plain Layout
 
-findPalindrome: Long => Long
-\end_layout
-
-\end_inset
-
- performing the following computation: First define 
-\begin_inset listings
-inline true
-status open
-
-\begin_layout Plain Layout
-
-f(n) = revDigits(n) + n
-\end_layout
-
-\end_inset
-
- for a given integer 
-\begin_inset listings
-inline true
-status open
-
-\begin_layout Plain Layout
-
-n
+f(n) = n + revDigits(n)
 \end_layout
 
 \end_inset
@@ -24214,39 +24203,38 @@ noprefix "false"
 \end_inset
 
 .
- If 
+ Using this 
 \begin_inset listings
 inline true
 status open
 
 \begin_layout Plain Layout
 
-f(n)
+f
 \end_layout
 
 \end_inset
 
- is a palindrome integer, 
+, write a function 
 \begin_inset listings
 inline true
 status open
 
 \begin_layout Plain Layout
 
-findPalindrome
+findPalindrome(n: Long): Long
 \end_layout
 
 \end_inset
 
- returns that integer.
- Otherwise, it keeps applying the same transformation and computes 
+ that returns the first palindrome integer found in the sequence 
 \begin_inset listings
 inline true
 status open
 
 \begin_layout Plain Layout
 
-f(n)
+n
 \end_layout
 
 \end_inset
@@ -24258,31 +24246,29 @@ status open
 
 \begin_layout Plain Layout
 
-f(f(n))
+f(n)
 \end_layout
 
 \end_inset
 
-, ..., until a palindrome integer is eventually found (this is mathematically
- guaranteed).
- A sample test:
+, 
 \begin_inset listings
-inline false
+inline true
 status open
 
 \begin_layout Plain Layout
 
-scala> findPalindrome(10101)
-\end_layout
-
-\begin_layout Plain Layout
-
-res0: Long = 10101
+f(f(n))
 \end_layout
 
-\begin_layout Plain Layout
+\end_inset
 
-\end_layout
+, etc.
+ (This sequence will often contain a palindrome integer, although it is
+ not mathematically guaranteed.) A sample test:
+\begin_inset listings
+inline false
+status open
 
 \begin_layout Plain Layout
 
@@ -24418,7 +24404,7 @@ scala> skip1(List(1, 2, 3, 4, 5))
 
 \begin_layout Plain Layout
 
-res0: List[Int] = List((1,3), (2,4), (3,5))
+res0: List[(Int, Int)] = List((1,3), (2,4), (3,5))
 \end_layout
 
 \end_inset
@@ -24481,7 +24467,33 @@ not
 \end_inset
 
 .
-
+ The required type signature and a sample test:
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+def largestNotThreeFiveSeven(n1: Long, n2: Long): Long = ???
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+scala> largestNotThreeFiveSeven(19, 97)
+\end_layout
+
+\begin_layout Plain Layout
+
+res0: Long = 96L
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -24498,6 +24510,33 @@ not
 \end_inset
 
  with the largest sum of decimal digits.
+ The required type signature and a sample test:
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+def largestSum(n1: Long, n2: Long): Long = ???
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+scala> largestSum(1, 54321)
+\end_layout
+
+\begin_layout Plain Layout
+
+res0: Long = 49999L
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -24586,7 +24625,34 @@ perfect numbers
 \end_inset
 
  is perfect.
- Determine all perfect numbers up to one million.
+ Determine all perfect numbers up to 10 thousand.
+ The required type signature and a sample test:
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+def perfectNumbersUpTo(n: Int): Seq[Int]  = ???
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+scala> perfectNumbersUpTo(10000)
+\end_layout
+
+\begin_layout Plain Layout
+
+res0: Seq[Int] = List(6, 28, 496, 8128)
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Subsubsection
@@ -26930,7 +26996,7 @@ res0: List[Int] = List(1, 2, 3, 4, 5, 6, 7, 8, 9)
 
 \begin_layout Plain Layout
 
-scala> iter.toList // Look at those elements again...??
+scala> iter.toList // Call the function again, get another result??
 \end_layout
 
 \begin_layout Plain Layout
@@ -27024,7 +27090,7 @@ This situation is impossible in mathematics: if
 \begin_inset Formula $f(x)+f(x)=20$
 \end_inset
 
- and obtain the correct result.
+..
  We could also set 
 \begin_inset Formula $y=f(x)$
 \end_inset
@@ -27074,6 +27140,7 @@ status open
 \begin_layout Plain Layout
 After applying a pure function, we can be sure that, for instance, no hidden
  values in memory have been modified.
+ - Not necessarily.
 \end_layout
 
 \end_inset
@@ -27208,7 +27275,7 @@ res0: List[Int] = List(1, 2, 3, 4, 5, 6, 7, 8, 9)
 
 \begin_layout Plain Layout
 
-scala> x.toList
+scala> x.toList   // Call the function again, get the same result.
 \end_layout
 
 \begin_layout Plain Layout
@@ -27302,7 +27369,7 @@ res0: List[Int] = List(1, 2, 3, 4, 5, 6, 7, 8, 9)
 
 \begin_layout Plain Layout
 
-scala> str.toList
+scala> str.toList   // Call the function again, get the same result.
 \end_layout
 
 \begin_layout Plain Layout
@@ -27363,7 +27430,7 @@ res0: List[Int] = List(1, 2, 3, 4, 5, 6, 7, 8, 9)
 
 \begin_layout Plain Layout
 
-scala> v.toList
+scala> v.toList   // Call the function again, get the same result.
 \end_layout
 
 \begin_layout Plain Layout
@@ -27741,7 +27808,8 @@ toStream
 \end_inset
 
  method.
- This restores the value-based reasoning because streams behave as values:
+ This recovers the correctness of value-based reasoning because streams
+ behave as values:
 \begin_inset listings
 inline false
 status open
@@ -27790,7 +27858,7 @@ res0: List[Int] = List(1, 2, 3, 4, 5, 6)
 
 \begin_layout Plain Layout
 
-scala> str.toList
+scala> str.toList   // Call the function again, get the same result.
 \end_layout
 
 \begin_layout Plain Layout