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
• z3_solve.py prove "induction_principle"

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
• z3_solve.py prove "soundness_theorem"

1. 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))"

Math_Verify

bash
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.