Proof Skeleton Generator

Generate structured proof skeletons with tactics, proof strategies, and key lemmas for theorems in Isabelle/HOL or Coq.

Workflow

1. Analyze the Theorem Statement

Examine the theorem to understand:

• Quantifiers: Universal (∀/forall) or existential (∃/exists)
• Logical structure: Implications, conjunctions, disjunctions
• Data types involved: Lists, natural numbers, custom types
• Complexity: Simple equality vs. complex property

2. Choose Target System

Ask the user which proof assistant to target:

• Isabelle/HOL: Uses Isar structured proofs, automatic tactics
• Coq: Uses Ltac tactics, more explicit proof terms
• Both: Generate skeletons for both systems

If not specified, default to generating both versions.

3. Determine Proof Strategy

Based on the theorem structure, identify the appropriate proof technique:

Induction - When theorem involves recursive types:

• List induction for list properties
• Natural number induction for arithmetic
• Structural induction for custom datatypes
• Strong induction when needed

Case Analysis - When theorem involves:

• Boolean conditions
• Option types (None/Some)
• Sum types or custom constructors
• Conditional expressions

Direct Proof - When theorem is:

• Simple equality that simplifies
• Follows directly from definitions
• Provable by automatic tactics

Forward Reasoning - Build up facts:

• Establish intermediate lemmas
• Chain implications
• Construct witnesses for existentials

Backward Reasoning - Work from goal:

• Apply rules to reduce goal
• Split conjunctions
• Introduce implications

4. Identify Required Lemmas

Determine helper lemmas that may be needed:

• Standard library lemmas: Check if already available
• Custom lemmas: Properties that need separate proof
• Induction hypotheses: How they'll be used
• Intermediate facts: Steps in the main proof

5. Generate Proof Skeleton

Use the reference files for tactics and patterns:

• Isabelle tactics: See isabelle_tactics.md
• Coq tactics: See coq_tactics.md
• Complete examples: See examples.md

Create a skeleton that includes:

1. Proof structure with appropriate method (induction, cases, etc.)
2. Case labels for each subgoal
3. Strategy comments explaining the approach
4. Tactic placeholders (sorry/admit) for incomplete steps
5. Intermediate assertions (have/assert) for key facts
6. Helper lemmas with their own skeletons

6. Structure the Output

Organize the proof skeleton clearly:

For Isabelle/HOL:

(* Helper lemmas if needed *)
lemma helper_name:
  "statement"
  sorry

(* Main theorem *)
theorem theorem_name:
  assumes "assumptions"
  shows "conclusion"
proof (method)
  case case_name
  (* Goal: ... *)
  (* Strategy: ... *)
  (* Key steps:
     1. ...
     2. ...
  *)
  show ?case sorry
next
  (* Additional cases *)
qed

For Coq:

(* Helper lemmas if needed *)
Lemma helper_name :
  statement.
Proof.
  (* proof *)
  admit.
Admitted.

(* Main theorem *)
Theorem theorem_name :
  statement.
Proof.
  intros.
  induction ... as [| ...].
  - (* Case: ... *)
    (* Strategy: ... *)
    admit.
  - (* Case: ... *)
    (* IH: ... *)
    (* Strategy: ... *)
    admit.
Admitted.

Key Principles

Clarity

• Add comments explaining the proof strategy
• Label each case clearly
• Document what the induction hypothesis provides
• Explain non-obvious steps

Completeness

• Include all necessary cases
• Identify all required helper lemmas
• Show the structure of nested proofs
• Indicate where automation might work

Practicality

• Use sorry (Isabelle) or admit (Coq) for incomplete steps
• Suggest automatic tactics where applicable
• Note when sledgehammer (Isabelle) might help
• Indicate which steps are trivial vs. challenging

Correctness

• Ensure case analysis is exhaustive
• Verify induction is on the right variable
• Check that the proof structure matches the goal
• Validate that lemmas are actually needed

Proof Strategy Selection Guide

When to Use Induction

List properties: forall xs, P xs

• Induction on xs
• Base case: empty list
• Inductive case: x :: xs with IH P xs

Natural number properties: forall n, P n

• Induction on n
• Base case: 0 or Suc 0
• Inductive case: Suc n with IH P n

Recursive function properties: When function is defined recursively

• Induct on the recursive argument
• IH mirrors the recursive call

When to Use Case Analysis

Conditional expressions: if b then ... else ...

• Split on b (true/false cases)

Option types: match opt with None => ... | Some x => ...

• Case on None and Some x

Sum types: match x with Left a => ... | Right b => ...

• Case on each constructor

When to Use Direct Proof

Definitional equalities: f x = g x where definitions unfold

• Unfold definitions and simplify

Trivial goals: Provable by auto, simp, reflexivity

• Try automatic tactics first

When to Use Lemmas

Repeated subgoals: Same property needed multiple times

• Extract as separate lemma

Complex intermediate facts: Multi-step derivations

• Prove as helper lemma

Standard properties: Check standard library first

• Import and use existing lemmas

Common Patterns

Induction with Simplification

proof (induction xs)
  case Nil
  show ?case by simp
next
  case (Cons x xs)
  show ?case using Cons.IH by simp
qed

Case Analysis with Subproofs

proof (cases x)
  case Constructor1
  show ?thesis
  proof -
    (* detailed steps *)
  qed
next
  case Constructor2
  (* ... *)
qed

Forward Reasoning Chain

proof -
  have step1: "fact1" by simp
  have step2: "fact2" using step1 by simp
  show ?thesis using step2 by simp
qed

Backward Reasoning with Rules

proof (rule some_rule)
  show "premise1" sorry
  show "premise2" sorry
qed

Tips

• Start with structure: Get the proof skeleton right before filling details
• Use comments liberally: Explain strategy and key steps
• Identify IH usage: Note where induction hypothesis is applied
• Check standard library: Many lemmas already exist
• Suggest automation: Note where auto, simp, blast might work
• Be realistic: Mark challenging steps as sorry/admit
• Both systems: Ensure semantic equivalence when generating both
• Nested induction: Handle carefully with clear variable names
• Termination: Note when termination proofs are needed