Air carrier pilot reported an engine failure due to contaminated oil. The aircraft was dispatched with an MEL regarding the 'Oil debris above limit'. ATC cleared the flight to divert.

Narrative

We were dispatched under MEL XX-XX-XXX due to the info message: 79 L ENGINE FAULT-OIL DEBRIS ABOVE LIMIT. Departure and the first 3 hours of flight were normal as far as the engine operation. At FL370 at XA26Z we received the: L ENG OPER DEGRADED message. I assigned the First Officer to continue flying and started the related Checklist. At XA27 we got: FMS PERF ACTIVE L ENG FAIL. I halted the previous Checklist after just a few steps; [requested priority handing] and told ATC we needed lower. Clearance was received. We accomplished the L ENG FAIL checklist followed by the L ENG Shutdown and FMS PERF Active checklist. Attempted APU start- no indications. Knowing there had been metal in the oil and that the auto relight failed we elected to not try a relight. We both agreed that ZZZ looked good given our location and the Thunderstorms near most major airports ahead. Arranged clearance to ZZZ with ATC and a right turn to avoid weather to our left. Notified infight (who had tried to call while we were running the shutdown procedure) code X. Notified the passengers. Notified Dispatch. While Captain was notifying above the First Officer accomplished reprogramming the destination for ZZZ while flying and communicating. Accomplished Non Normal Landing Data Calc checklist (no wifi and no ACARS aerodata; done manually). Both pilots set up for visual with ILS backup for XXR in ZZZ. Captain reviewed for First Officer all checklists accomplished and limitations/considerations. First Officer briefed approach. All above and Approach checklist accomplished by mid teens ALT. Normal approach and landing; CFR standing by; taxied to Gate with no assistance. Did Logbook writeup.