Each step in our proof is a formula but not only that

Each step in our proof is a formula, but not only that; the entire proof is also a
formula! The proof is a predicate. Written out in the usual way it is
distance = velocity * time = 60 * time = 60 * 4 = 240
Without the annotations, our proof is just an ordinary Z predicate (it uses the Z
convention that a = b = c is an abbreviation for a = b A b = c).

15.1. Calculation and proof 151
A proof can be any predicate; it need not be an equation. Let's change the problem
statement slightly to say that the train is moving at less than sixty miles per hour:
velocity < 60. Then the proof becomes:
distance = velocity * time [Definition]
< 60 * time [velocity < 60]
= 60 * 4 [time = 4]
= 240 [Arithmetic]
Written out on one line, this is
distance = velocity * time < 60 * time = 60 * 4 = 240
This time we obtain the inequality distance < 240 because we can infer a < d
from a = b < c = d.
Exercise 15.1.1 Equality =, greater than >, and less than < are all transitive. Is
inequality ^ transitive as well?
Calculations need not be arithmetic. This example uses set membership G and the
subset relation c. Organizations are modelled as sets of people, and organizational
hierarchy is modelled as subset relations. Philip works on the adhesives team in the
materials group, which is part of the research division.
philip : PERSON
adhesives, materials, research, manufacturing : ¥ PERSON
adhesives c materials
materials c research
philip 6 adhesives
Intuition tells us that Philip must work in the research division. We don't have to
write a formula to say that because it is easy to show:
philip e adhesives [Definition]
c materials [Definition]
c research [Definition]
This is a formal proof of the predicate philip e research. It uses the transitivity of
the subset relation: From S c T C U we can infer 5 c U. The proof is trivial but it
can save us a great deal of writing. Without it we would have to include the predicates

152 Chapter 15. Formal reasoning
philip € materials and philip € manufacturing in the definition — and so on for
every other employee. Now that we can do the proof, we don't have to include these
facts; we can infer them when they are needed. If the company is reorganized — for
example, materials is moved from research to manufacturing — we only need
to change a single line.
In the preceding proofs, each line shows an expression which is joined to the
expression on the preceding line by an equal sign or a relation symbol. We can
also build proofs where the lines are predicates joined by logical connectives. The
connective is often equivalence, which plays much the same role for predicates that
equality does for expressions. Here is a formalization of the little algebra problem:
find x, given 2x + 7 = 13.
x:Z
2*x + 7= 13
We simply solve for x
2 * x + 7 = 13 [Definition.]
<»2*JC = 1 3 - 7
[Subtract 7 from both sides.]
4» 2 * x = 6 [Arithmetic]
=>• (2 * x) div 2 = 6 div 2 [Divide both sides by 2.]
^ x = 6 div 2 [Division on left side, algebra]
O x — 3 [Division on right side, arithmetic]
This completes our proof of the predicate 2*;t + 7=13=^;t = 3.