[Verse 1]
When the proof seems hard to find
And the path is undefined
There's another way to go
Let me teach you what I know
Instead of climbing straight ahead
Take the contrapositive thread
Flip the arrow, flip each side
Let the logic be your guide
[Chorus]
When you can't prove P implies Q
Try not Q implies not P too
Same destination, different road
Going around lifts the heavy load
Contrapositive, contrapositive
Makes the impossible possible
Contrapositive, contrapositive
When direct proof's not workable
[Verse 2]
Take the case of even squares
Direct proof brings lots of cares
If n squared is even, then n's even
But the forward path's not given
Flip it round and see the light
If n is odd, then prove the sight
That n squared must be odd as well
Now the structure starts to tell
[Chorus]
When you can't prove P implies Q
Try not Q implies not P too
Same destination, different road
Going around lifts the heavy load
Contrapositive, contrapositive
Makes the impossible possible
Contrapositive, contrapositive
When direct proof's not workable
[Bridge]
Let n equal two k plus one
Now the algebra's begun
Square it out and you will see
Four k squared plus four k plus one
Factor out the two and find
Two times something plus one's the sign
Of an odd number standing tall
Contrapositive proves it all
[Verse 3]
When existence is your goal
But you can't control the whole
Flip to universal truth
That's your mathematical proof
Concrete structure in your hand
Helps you understand and plan
Round the mountain, not on top
Same arrival, better stop
[Chorus]
When you can't prove P implies Q
Try not Q implies not P too
Same destination, different road
Going around lifts the heavy load
Contrapositive, contrapositive
Makes the impossible possible
Contrapositive, contrapositive
When direct proof's not workable
[Outro]
Flip the arrow, flip each side
Let contrapositive be your guide
Same truth, easier way
Proof by contrapositive saves the day