Vibecosystem proof-theory

Problem-solving strategies for proof theory in mathematical logic

install

source · Clone the upstream repo

git clone https://github.com/vibeeval/vibecosystem

Claude Code · Install into ~/.claude/skills/

T=$(mktemp -d) && git clone --depth=1 https://github.com/vibeeval/vibecosystem "$T" && mkdir -p ~/.claude/skills && cp -r "$T/skills/math/mathematical-logic/proof-theory" ~/.claude/skills/vibeeval-vibecosystem-proof-theory && rm -rf "$T"

manifest: skills/math/mathematical-logic/proof-theory/SKILL.md

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.