1. DemeterF

2. Illustration of combine for capacity constraint violation19 :C
(w1+w2+w3+w4,2)
:Cons
(w1+w2+w3+w4,1) :E
(w4,0) 1 20 2
18a 3 18 3a
13a
:Cons
(w1+w2+w3,1)
after blue arrow
combine is
active (like after)
17a :C
(w2+w3,1) 4
12a 17
:Cons
(w1,0) 5 13 14 12
16a
:Cons
(w2+w3,0)
15a 6 :E
(w1,0)
:Empty
(0,0)
11a 7a 15 16 11 :E
(w3,0)
:Cons
(w2,0) 8
both containers (C) violate
capacity constraints 7
10a 9a :E
(w2,0)
:Empty
(0,0) 9 10

3. The default Builder for PathSpec
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

4. well-formed movie
show how the default builder is modified to combine Boolean objects.

5. The default Builder for PathSpec well-formed specialization 1
#t: true
#f: false
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J #t :J 5 2 :S :S :S :S 3 #t 7 #f 8 #t 4 #t

6. The default Builder for PathSpec well-formed specialization 2
#t: true
#f: false
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J #t :J #t 5 2 :S :S :S :S #t 7 #f 8 #t 3 4 #t

7. The default Builder for PathSpec well-formed specialization 3
#t: true
#f: false
after blue arrow
copy is built
(like after) #t 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 3 #t 7 #f 8 #t 4 #t

8. The default Builder for PathSpec well-formed specialization 4
#t: true
#f: false
after blue arrow
copy is built
(like after) #t 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. #f 6 :J :J 5 2 :S :S :S :S 3 #t 7 #f 8 #t 4 #t

9. The default Builder for PathSpec well-formed specialization 5
#t: true
#f: false
after blue arrow
copy is built
(like after) #t 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. #f 6 :J #t :J 5 2 :S :S :S :S #t 7 #f 8 #t 3 4 #t

10. The default Builder for PathSpec well-formed specialization 6
#t: true
#f: false
after blue arrow
copy is built
(like after) #t 10 #f :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 3 #t 7 #f 8 #t 4 #t

11. The default Builder for PathSpec well-formed specialization 7
#t: true
#f: false
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 3 #t 7 #f 8 #t 4 #t

12. The default Builder for PathSpec well-formed specialization
#t: true
#f: false
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 #f 8 #t 3 #t 4 #t

13. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

14. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

15. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

16. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

17. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

18. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

19. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

20. The default Builder for PathSpec NOT_JOIN_MERGE specialization
after blue arrow
copy is built
(like after) 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

21. The default Builder for PathSpec NOT_JOIN_MERGE specialization
insert NOT
after blue arrow
copy is built
(like after) :N 10 :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

22. The default Builder for PathSpec NOT_JOIN_MERGE specialization
insert NOT :M
after blue arrow
copy is built
(like after) :N 10 :N :M 1 9
Count only upon
first visit (red) and
upon final visit (blue).
For leaf nodes, count
only in red. 6 :J :J 5 2 :S :S :S :S 7 8 3 4

23. Illustration of combine for capacity constraint violation
(w1+w2+w3+w4,2)
:Cons
(w1+w2+w3+w4,1) :E
(w4,0)
:Cons
(w1+w2+w3,1) :C
(w2+w3,1)
:Cons
(w1,0)
:Cons
(w2+w3,0) :E
(w1,0)
:Empty
(0,0) :E
(w3,0)
:Cons
(w2,0)
both containers (C) violate
capacity constraints :E
(w2,0)
:Empty
(0,0)

24. Theory t[f,b](d) => d’, where d’=f(d), d is atomic
On left side of => the term c(…) only indicates a compound object.
Theory
t[f,b](d) => d’, where d’=f(d), d is atomic
t[f,b](c(d0, … ,dn)) =>
f(b(c(d0, … ,dn), d’0, … ,d’n)), where d’i = t[f,b](di)
Default functions:
id[f](d) => d
id[b](c(d0, … ,dn), d’0, … ,d’n) => c(d0, … ,dn)
b[c](c(d0, … ,dn), d’0, … ,d’n) => c(d’0, … ,d’n)

25. Theory
f is a polymorphic function that takes a single argument and returns a result.
b is a function object that is responsible for reconstruction of data types.