TAS increases with less dense air, which is intuitive. What I don't understand is why IAS decreases.
This is assuming sub-transonic speeds where compressibility effects are negligible.
To phrase this in more practical terms, if you look at the climb table in a POH for a turbocharged airplane you will see full throttle climb rate drops with an increase in altitude/temp, even before the MP drops. EAS and propeller efficiency would not seem sufficient to explain the IAS drop seen.
I would think that given a TAS, pressure altitude, and temperature, one could derive the IAS but I have not seen any math to do this.
This is in reference to a turbocharged propeller driven airplane that is capable of maintaining power at higher altitudes.