Backwards search
The sext.w removal pass (before the new patch) checks if the input to sext.w is already in sign-extended form, so it can eliminate it. It does that by checking every definition/source that reaches the sext.w is an instruction that produces a sign-extended value, either by definition (e.g. ADDW), or it propagates sign-extension (e.g. OR) so we check its sources recursively.
Forward search
Sometimes, one of the sources is an instruction that doesn't always produce a sign-extended value, but it has a W-version that does (e.g. ADD / ADDW). If we transform the ADD to ADDW, the sext.w can be removed (assuming other def paths are satisfied), but this transformation is sound only if every use of this ADD/W only reqruires the lower 32-bits either directly (like sll %x, 32) or they propagate dependency (lower word of output only depends on lower word of input) so we check its uses recursively.
When searching backwards, if an instruction that can be replaced with W-variant is encountered, this pass runs the forward search to verify it can be replaced, then adds it to a list of fixable instructions. After verifying all paths, it replaces the instruction and removes the sext.w.
Drop else after return.