[Kutta]

A type system cannot be TC. What you seem to talk about, is that if a type checker can simulate arbitrary TM's through an encoding of its input, then the type checker is necessarily non-total. This is correct. But the Idris checker is total and it is not possible to use it to simulate TMs.

The Idris type system does allow you to specify and formalize TMs internally, but that has no bearing on decidability of type checking, as I explained before.

[potiuper]

A type system can be TC. A type system is a logical system comprising a set of rules that assigns a property called a type to the various constructs of a computer program, such as variables, expressions, functions or modules. If those rules can encode a Turing complete system, then the system is Turing complete. Type checking is the process of verifying and enforcing the constraints of types. Type assignment is different from verification.