proof-theory

# 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/cc_math/z3_solve.py prove "P(0)" ``` ### Z3_Induction_Step ```bash uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "ForAll([n], Implies(P(n), P(n+1)))" ``` ### Z3_Soundness ```bash uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "Implies(derivable(phi), valid(phi))" ``` ...