Proof Theory

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
   • Completeness: if valid then provable
   • z3_solve.py prove "soundness_theorem"

5. Proof Verification

   • Check each step follows from rules
   • Verify dependencies are satisfied
   • math_scratchpad.py verify "proof_steps"

Tool Commands

Z3_Induction_Base

uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "P(0)"

Z3_Induction_Step

uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "ForAll([n], Implies(P(n), P(n+1)))"

Z3_Soundness

uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "Implies(derivable(phi), valid(phi))"

Math_Verify

uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py verify "proof_structure"

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.