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

1. Structural Induction

- Define well-founded ordering on structures - Base: prove for minimal elements - Step: assume for smaller, prove for current - z3solve.py prove "inductionprinciple"

1. Cut Elimination

- Gentzen's Hauptsatz: cuts can be eliminated - Subformula property: only subformulas appear - Useful for proof normalization

1. Completeness/Soundness Check

- Soundness: if provable then valid - Completeness: if valid then provable - z3solve.py prove "soundnesstheorem"

1. Proof Verification

- Check each step follows from rules - Verify dependencies are satisfied - mathscratchpad.py verify "proofsteps"

Tool Commands

Z3InductionBase

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

Z3InductionStep

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 .maestro/skills/math-mode/SKILL.md for full tool documentation.