Proof Theory

When to Use

Use this skill when working on proof-theory problems in mathematical logic.

Decision Tree

1. Proof Strategy Selection

   • Direct proof: assume premises, derive conclusion
   • Proof by contradiction: assume negation, derive false
   • Proof by cases: split on disjunction
   • Induction: base case + inductive step

2. Structural Induction

   • Define well-founded ordering on structures
   • Base: prove for minimal elements
   • Step: assume for smaller, prove for current
   • z3_solve.py prove "induction_principle"

3. Cut Elimination

   • Gentzen's Hauptsatz: cuts can be eliminated
   • Subformula property: only subformulas appear
   • Useful for proof normalization

4. Completeness/Soundness Check

   • Soundness: if provable then valid