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vibeeval / proof-theory

Problem-solving strategies for proof theory in mathematical logic

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SKILL.md
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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

Z3InductionBase

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

Z3InductionStep

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.