Basic Proof techniques

So far, all the techniques I used in my proof are induction proof and modus ponens, and other propositional judgements. I really think that I can do my proof on the theorem prover, that will definitely eliminate my doubts of correctness of my proof.

Theory and Applications

Sometimes I couldn't help to wonder: Am I a theorist or a practitionist? Or should we distinct these two as two different objects,or even think these two are on the each side of a polar. And I sense that there is a trend that one does not respect the other. I realize at least I have this kind of prone. And I also notice that most theorists pay no attention to the application, and the practitionists do the same things. I think for any subjects, there are three kinds of people, one is pure theorist, one is trying to put the theory he understands to the practice, one focus only on practice, he assume the existence of a theory foundation for his work, all he need to do is improved the practical impact of it. If this kind of situation is true, I feel that my favor is somewhere between the first two, or I want to be both the first two kind of people at the same time. yes, I want to invent my own theory, and see how it work out in practice. And the sadness is some theorist cannot see the flourish of his theory in their lifetime. But, I want to be an optimist, and with the hope and passion, to delve into the pure theory,figure out the essential relations of the objects.

The world

To see a world in a grain of sand
Or a heaven in a wild flower,
Hold infinity in the palm of your hand
And eternity in an hour.
WILLIAM BLAKE (1757-1827)

Thinking

CLAUDE SHANNON once told me that as a kid, he remembered being stuck on a jigsaw puzzle.
His brother, who was passing by, said to him:
"You know: I could tell you something."
That's all his brother said.
Yet that was enough hint to help Claude solve the puzzle.
The great thing about this hint... is that you can always give it to yourself !!!
I advise you, when you're stuck on a hard problem,
to imagine a little birdie or an older version of yourself whispering
"... I could tell you something..."

Cited from http://www-2.cs.cmu.edu/~mblum/research/pdf/grad.html

Stand in the Truth of Mountain And see the Chaos Under

After A month's Studying and learning and Researching, I finally see through what's the structure behind the lambda D system. That's fantastic! Amazingly, the structure is so simple that I feel lambda D actually only have a little tiny improvement compare to simple typed lambda system. Anyway, Nail it!

Thoughts about Proof by Induction.

When we use the method of Proof by Induction,
First, we need to figure out we do induction on what.
Second, when we figure out the thing that we're going to do induction on, we also
need to show the well-foundness of that thing.
Finally, we can safely use the Induction Method.

Different Opinions

People use different point of view to see their research, some use practical point of view, some just don't. Can we say one is better than the other? I don't think they are comparable. But sadly sometimes there are people try to compare them and try to convince me one is better than the other, what should I say.