majiayu000 Proof Theory
When to Use
Use this skill when working on proof-theory problems in mathematical logic.
Decision Tree
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
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"
Cut Elimination
- Gentzen's Hauptsatz: cuts can be eliminated
- Subformula property: only subformulas appear
- Useful for proof normalization
Completeness/Soundness Check
- Soundness: if provable then valid
- Completeness: if valid then provable
z3_solve.py prove "soundness_theorem"
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/z3_solve.py prove "P(0)"
Z3_Induction_Step
uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll([n], Implies(P(n), P(n+1)))"
Z3_Soundness
uv run python -m runtime.harness scripts/z3_solve.py prove "Implies(derivable(phi), valid(phi))"
Math_Verify
uv run python -m runtime.harness scripts/math_scratchpad.py verify "proof_structure"
Cognitive Tools Reference
See .claude/skills/math-mode/SKILL.md for full tool documentation.