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/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.