Ensuring non-decreasing lagged differences

In the data example below, the cumulative values (cumvals) should ideally be non-decreasing (equivalently the consecutive differences (diffs) must be non-negative). However, in some instances, that's not the case due to revisions that result to data updates at a later date. When such revisions are made, changes need to be made so that the cumulative values are monotonically non-decreasing. I've attempted fixing this issue for a while without success.

cumvals <- c(0, 0, 1, 0, 1, 1, 2, 5, 7, 7, 6, 5, 8, 8, 7)
ID <- rep(1001, length(cumvals))
df <- data.frame(ID, cumvals)
  mutate(diffs= c(0, diff(cumvals)))

The corrected values should be as in the data frame below.

Ideally, if the trend for cumvals decreases from say day 3 to day 4, we want to keep the smaller value for day 4 and modify the value for day 1 to equal to that for day 2 and preserve the non-decreasing aspect. Now, if the value for days 2 and 3 are similar, then we have a decrease at day 4, we need to modify the values for days 2 and 3 and assign them to value for day 4. The last scenario is when their are consecutive decreases. For example, when we have consecutive instances of 7, 7, 6, 5, we want modify this to 5, 5, 5, 5. That way the vector/sequence is non-decreasing.

corrcumvals <- c(0, 0, 0, 0, 1, 1, 2, 5, 5, 5, 5, 5, 7, 7, 7)
df2 <- data.frame(ID, cumvals, corrcumvals)
  mutate(diffs= c(0, diff(cumvals)), corrdiffs= c(0, diff(corrcumvals)))

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