Coq中无需预先定义Theorem的交互式定理证明最优方案问询

Great question! Coq absolutely supports this kind of open-ended, exploratory workflow where you want to experiment with assumptions, definitions, and proof transformations without locking yourself into a predefined theorem upfront. Here's how to set it up and make the most of Coq's interactive tools:

1. Use Sections to Encapsulate Your Context #

First, wrap your initial assumptions and definitions in a Section—this keeps them contained (so they don't pollute your global namespace) and lets you reuse them across multiple exploratory proofs:

Section NumberTheoryExploration.
  (* Define base types and assumptions *)
  Variable n m : nat.
  Hypothesis even_n : exists k, n = 2 * k.
  Hypothesis even_m : exists k, m = 2 * k.

  (* Add helper definitions as you go *)
  Definition sum := n + m.

2. Launch Interactive Proofs with Goal #

Instead of declaring a Theorem or Lemma upfront, use the Goal command to start an interactive proof session with any statement you want to explore. Coq will drop you into its standard interactive mode, tracking your hypotheses and current target just like it would for a named theorem:

Goal even sum.

Now you can use all your favorite tactics (intros, rewrite, destruct, simpl, etc.) to manipulate the goal and hypotheses. At any point, run:

• Show to view the current proof state (hypotheses + target)
• Show Hints to get tactic suggestions relevant to your current state
• Check <term> to verify the type or value of any term in your context

3. Iterate and Save Discoveries #

• If you want to abandon a line of exploration and try a new goal, run Abort to exit the current proof session cleanly.
• When you stumble on a meaningful result you want to keep, use Save <TheoremName> to turn your current proof into a named theorem:

  Save EvenSumOfEvens.
  

  Or if you've finished the proof, Qed will finalize it just like with a predeclared theorem.

4. Bonus Tips for Exploration #

• Use Eval compute in <term> to test the behavior of your definitions (e.g., Eval compute in sum.) and validate transformations.
• Enable Set Printing All. to see full, unshortened terms in your proof state—this helps debug confusing rewrites or hypothesis changes.
• Use Print Context. to list all variables, hypotheses, and definitions active in your current section.

Does Coq Support This Workflow? #

Absolutely! Coq's interactive proof engine is designed to track context and goal state dynamically, regardless of whether you started with a predeclared theorem or a temporary Goal. This exploratory approach is actually common among researchers and developers who want to formalize mathematical ideas without knowing the exact end result upfront.

Once you're done with your exploration, you can close the section with End NumberTheoryExploration. to clean up the context.

内容的提问来源于stack exchange，提问作者user9335697