If rules are too difficult just call them facts.
We define counting on by one and give it the special prime symbol.
Counting on Facts
We define addition by two facts for counting on.
Counting on by zero fact
n+0 = n
Counting on by one more fact
m+n’ = (m+n)’
This fact says that counting on by one more of the right addend equals counting on by one more from the previous sum.
Examples of counting on facts used to define addition
2+0 = 2
Suppose we have already done
2+3 = 5.
Now we want to do 2+4. Here 4 = 3′ we also know.
2+4 = 2+3′ = (2+3)’ = 5′ = 6
This is how we use that 2+3 = 5 in the substitution step.
If defining addition is too hard, call it finding addition facts.