[Verse 1] In the realm of proof we start our quest Show me something real exists at first Constructive path means point and say "Here's your object on display" Like perfect six with one plus two plus three Exhibit clearly what you need to see [Chorus] Existence first, then uniqueness too Show at least one, then at most one's true Build your proof in stages clean Demonstrate what can't be seen If two objects share the trait Prove they must be duplicate [Verse 2] Non-constructive takes a twisted road Assume nothing fits your proposed code Let contradiction rear its head When non-existence leads to dread No need to name the hidden prize Just show denial tells you lies [Chorus] Existence first, then uniqueness too Show at least one, then at most one's true Build your proof in stages clean Demonstrate what can't be seen If two objects share the trait Prove they must be duplicate [Verse 3] For uniqueness take this classic route Suppose both a and b compute The property you're testing for Then show they're equal at the core If two things satisfy your claim They must in fact be just the same [Bridge] Division algorithm shows the way Any integer a and b where b's not zero Quotient q and remainder r appear With a equals b times q plus r And zero less than or equals r less than b Exists unique, as proved theoretically [Chorus] Existence first, then uniqueness too Show at least one, then at most one's true Build your proof in stages clean Demonstrate what can't be seen If two objects share the trait Prove they must be duplicate [Outro] Constructive or by contradiction Uniqueness needs your clear conviction Mathematics builds on solid ground When existence and uniqueness are found
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