ggplot2 plotting a 100% stacked area chart

I want to plot a 100% stacked area chart out of a dataframe, where each row of the df sums up to 1. An example dataframe is here: https://pastebin.com/ADMQP6Nx

What i want ultimately is something like this:

Most solutions i found so far here on SO have a different data.frame structure, where the grouping variables are defined in rows rather than as columns, e.g.:

+--------+-------+-------+
| Period | Group | Value |
+--------+-------+-------+
| 1      | t1    | 0.3   |
| 1      | t2    | 0.1   |
| 1      | t3    | 0.4   |
| 1      | t4    | 0.2   |
| 2      | t1    | 0.5   |
| 2      | t2    | 0.1   |
| 2      | t3    | 0.3   |
| 2      | t4    | 0.2   |
| ...    | ...   | ...   |
+--------+-------+-------+

And they use then ggplot like this:

ggplot(data, aes(x=Period, y=Value, fill=Group)) +  geom_area()

Is there a solution without transforming the data frame?

Transforming the data with gather:

library(tidyverse)

df <- structure(
  list(
    Period = 1:4,
    t1 = c(0.3, 0.5, 0.1, 0.4),
    t2 = c(0.1, 0.1, 0.3,
           0.2),
    t3 = c(0.4, 0.2, 0.4, 0.3),
    t4 = c(0.2, 0.2, 0.2, 0.1)
  ),
  .Names = c("Period", "t1",
             "t2", "t3", "t4"),
  row.names = c(NA, -4L),
  class = "data.frame"
)


df %>% 
  gather(Group, Value, t1:t4) %>% 
  ggplot(aes(x=Period, y=Value, fill=Group)) + 
  geom_area()