Useful proof principles and strategies
In teaching students about proofs and their development, it helps to
be able to demonstrate principles and strategies for developing proofs.
Here are several useful ones that appear in
A Logical Approach. They are perhaps obvious to the mature
mathematician or computer scientist. However, to students they are
not. Mathematicians and computer scientists have, in general, not
discussed such principles and strategies, and that is one reason
students have previously had trouble developing proofs.
- Heuristic. To prove P == Q, transform P == Q to a known
theorem, transform P to Q, or transform Q to P.
- Heuristic. The operators that appear in an expression
and the shape of its subexpressions
focus the choice of theorems to be used in manipulating it.
Therefore, in developing the next step of a proof, identify applicable theorems
by matching theorems with the structure of the subexpressions of the current
expression.
- Heuristic of Operator Elimination. To prove a theorem about an operator,
first eliminate the operator using its definition, then manipulate, and finally
reintroduce the operator (if necessary).
- Heuristic. To prove P == Q, transform the side with the most
structure (either P or Q) into the other.
- Principle. Structure proofs to minimize the number of rabbits pulled
out of a hat --make each step seem obvious, based on the structure of
the expression and the goal of manipulation.
- Principle. Lemmas can provide structure, bring to light interesting
facts, and ultimately shorten a proof.
- Heuristic. Exploit the ability to parse theorems like Golden rule,
p /\ q == p == q == p \/ q, in many different ways (using
associativity and symmetry transparently).
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