---
name: proof-theory
description: "Problem-solving strategies for proof theory in mathematical logic"
allowed-tools: [Bash, Read]
---
# 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
```bash
uv run python -m runtime.harness scripts/z3_solve.py prove "P(0)"
```
### Z3_Induction_Step
```bash
uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll([n], Implies(P(n), P(n+1)))"
```
### Z3_Soundness
```bash
uv run python -m runtime.harness scripts/z3_solve.py prove "Implies(derivable(phi), valid(phi))"
```
### Math_Verify
```bash
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.
