Recent proposals for trusted hardware platforms, such as Intel SGX and the MIT Sanctum processor, offer compelling security features but lack formal guarantees. We introduce a verification methodology based on a trusted abstract platform (TAP), a formalization of idealized enclave platforms along with a parameterized adversary. We also formalize the notion of secure remote execution and present machine-checked proofs showing that the TAP satisfies the three key security properties that entail secure remote execution: Integrity, confidentiality and secure measurement.We then present machine-checked proofs showing that SGX and Sanctum are refinements of the TAP under certain parameterizations of the adversary, demonstrating that these systems implement secure enclaves for the stated adversary models.