  We theoretically study superconducting proximity effect between a topological
insulator (TI) and a high-temperature d-wave superconductor (dSC). When the
TI-dSC heterostructure violates 90 degree-rotation and certain reflection
symmetries, we show that a sizable s-wave pairing, coexisting with a d-wave
one, emerges in the proximity-induced superconductivity in the TI's top surface
states. Weak disorder further suppresses d-wave pairing but not s-wave one in
the TI's surface states. More importantly, the pairing gap in surface states is
found to be nodeless and nearly-isotropic when the Fermi pocket of surface
states is relatively small. Our theoretical results qualitatively explain
recent experimental evidences of a nearly-isotropic pairing gap on surface
states of Bi_2Se_3 induced by proximity with high-T_c cuprate
Bi_2Ca_2Cu_2O_{8+\delta}. We also demonstrate convincing evidences of Majorana
zero modes in a magnetic hc/2e vortex core, which may be detectable in future
experiments.
