In reply to John_Hat:
Not strictly the same thing but had an interesting nylon-on-nylon melting experience back in 1996, although not rope-on-rope; instead rope-on-Whillans Harness. I think the principle is probably similar though.
We were at the top of The Wand on Creag Meaghaidh and my climbing partner was leading through on the final easy slopes to the summit plateau. She'd nearly reached horizontal ground and was almost a rope length out when the whole slope avalanched. Carried down at speed she picked up momentum and became airborne, flying over the top of my head as the avalanche poured over and around the belay, and she fell back down the route like a bungee jumper. Thankfully the belay held and she was unscathed and able to climb back up and we got off the route second go.
The tie-in knot in my old Whillans had been cinched so tight (or so I thought) that I couldn't actually get it undone so just took the harness off and stuffed it and the rope in the 'sac and we went home.
It was only the next morning on closer inspection that I discovered why the tie-in was so difficult to undo - the rope had melted due to intense friction and hidden beneath the Whillans nylon tie-in loops there were only two strands of inner core left intact.
That was all that had stopped me from detaching from the belay and following her on a one-way journey to the bottom.
I used to happily and recklessly climb on a single 9 mm rope in winter quite a lot in those days. Needless to say, although melting ropes are a rare occurrence, as has been pointed out by others, I've always used two ropes since.
And retired the Whillans Harness...