Which doctrine was formalised first: Infant Baptism or Original Sin?

What I mean is something like this :
**Case A, Original Sin came first
** because all newly born baby are going to hell if they die in their infant age
then let's do Infant Baptism so they don't go to hell if the infant die.
(here, Infant Baptism start to exist after the Original Sin formulated). **Case B, Infant Baptism came first
** because the church do Infant Baptism
Then it must be because all newly born baby will go to hell if the infant die.
(here, Original Sin formulated because the church do Infant Baptism). Here is what I read from this link > Pelagius promised infants who died unbaptised entry into “eternal > life” (not, however, into the “Kingdom of God” [Jn 3:5]), reasoning > that God would not condemn to hell those who were not personally > guilty of sin From the same link > In countering Pelagius, Augustine was led to state that infants who > die without Baptism are consigned to hell ---//cut--- **Liturgical practice confirms the Church's belief that all inherit Adam's sin and must be transferred from the power of darkness into the kingdom of light** From reading those sentences (especially the bold one), my own conclusion is case-B.
So, there is already an Infant Baptism ritual in that time. Maybe Pelagius say *"You don't need to baptize infant. If they die in their infant age, they won't go to hell"*. On the other hand, Augustine do not agree on what Pelagius say. In the pov of Augustine, Infant Baptism must be performed in order that if the infant die, the infant won't go to hell. But since I'm not so sure, that's why I ask here.