Relation between Interest Rates and Put value

CFA Level II Derivatives
#1

I know that the curriculum states that interest rates and put value are negatively related.

There’s probably something missing in my logic, but suppose I have an embedded put in my bond. Thus:

Put = Putable bond - Straight bond

Put = (Straight bond + Put) - Straight bond

Put = [Straight bond + (Exercise price - Straight bond)] - Straight bond

Put = Exercise price - Straight bond

If interest rates goes up → Straight bond would go down → Then Put value would go up?

#2

I suspect that they’re talking about the value of a put option on a stock, not a put embedded in a putable bond.

Did you get this statement from a discussion of put-call parity?