AgentSkillsCN

proof-theory

数理逻辑中证明论的解题策略

SKILL.md
--- frontmatter
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

  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/z3_solve.py prove "P(0)"

Z3_Induction_Step

bash
uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll([n], Implies(P(n), P(n+1)))"

Z3_Soundness

bash
uv run python -m runtime.harness scripts/z3_solve.py prove "Implies(derivable(phi), valid(phi))"

Math_Verify

bash
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.