| name | proof-theory |
|---|---|
| description | Problem-solving strategies for proof theory in mathematical logic |
| allowed-tools | ["Bash", "Read"] |
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.