Extend semi-automatic proof search to the full dependent proof tree type

Develop semi-automatic proof search techniques for the complete dependent type proofTreeOf Ps j, lifting the current approach that works only for a simplified proof tree type to the full rule set with evidence-dependent judgments.

Background

The paper presents a semi-automated approach for constructing proof trees but only for a simplified version of the proof tree type that omits evidence and assumptions. This simplification enables some automation but does not cover the full logic as mechanized elsewhere in the paper.

The authors explicitly state that bringing the same level of automation to the full dependent proofTreeOf Ps j type (which includes evidence and assumptions) is deferred, marking a concrete extension challenge.