A pass that changes the tree is a rewrite. The rewrite is a rule with a shape on the left and a shape on the right, and the shapes are what make it provable.
About 1 min of reading
Fold and prove equality
Write the rule
Before you run it, say out loud what it will print. Then open the answer.
Run to a fixed point
Before you run it, say out loud what it will print. Then open the answer.
Write the bound
Before you run it, say out loud what it will print. Then open the answer.
One pass can open a new match. Running to a fixed point is running until the list is empty, and empty is a checkable answer.
A maximum number of rounds. The number is the bound, and the bound is what turns a loop into a decision.
The left shape is matched, the right shape replaces it. Two shapes is a rule, and a rule is a line in a file.
The answer before equals the answer after. The equality is the claim, and the claim is checked by a test.
A rule whose right side matches its left side runs forever. The rule is refused at load, and the refusal is a name.
Each one is defined here on this site. Open the entry for the full definition and the other words it points at.