ANTPG/annotated-isabelle

Extensively annotated Isabelle source code, suitable for pretraining, about 500M tokens in Qwen3's tokenizer.

Data source: Isabelle/HOL + AFP 2025-02-12. Cases from the PISA benchmark are removed.

Example:

lemma (in group) diff_neutralizes: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "subgroup H G" "R ∈ rcosets H"
  shows "⋀r1 r2. ⟦ r1 ∈ R; r2 ∈ R ⟧ ⟹ r1 ⊗ (inv r2) ∈ H"
proof (-)
  (*goal: ‹⋀r1 r2. ⟦r1 ∈ R; r2 ∈ R⟧ ⟹ r1 ⊗ inv r2 ∈ H›*)
  fix r1 and r2
  assume r1: "r1 ∈ R" and r2: "r2 ∈ R" (*‹(r1::'a) ∈ (R::'a set)› ‹(r2::'a) ∈ (R::'a set)›*)
  obtain g where g: "g ∈ carrier G" "R = H #> g"
    (*goal: ‹(⋀g. ⟦g ∈ carrier G; R = H #> g⟧ ⟹ thesis) ⟹ thesis›*)
    using assms (*‹subgroup (H::'a::type set) G› ‹R ∈ rcosets H›*) unfolding RCOSETS_def
    (*goal: ‹(⋀g. ⟦g ∈ carrier G; R = H #> g⟧ ⟹ thesis) ⟹ thesis›*)
    by blast
  then obtain h1 and h2 where h1: "h1 ∈ H" "r1 = h1 ⊗ g" and h2: "h2 ∈ H" "r2 = h2 ⊗ g"
    (*goal: ‹(⋀h1 h2. ⟦h1 ∈ H; r1 = h1 ⊗ g; h2 ∈ H; r2 = h2 ⊗ g⟧ ⟹ thesis) ⟹ thesis›*)
    using r1 (*‹(r1::'a) ∈ (R::'a set)›*) r2 (*‹r2 ∈ R›*) unfolding r_coset_def
    (*goal: ‹(⋀h1 h2. ⟦h1 ∈ H; r1 = h1 ⊗ g; h2 ∈ H; r2 = h2 ⊗ g⟧ ⟹ thesis) ⟹ thesis›*)
    by blast
  hence "r1 ⊗ (inv r2) = (h1 ⊗ g) ⊗ ((inv g) ⊗ (inv h2))"
    using inv_mult_group (*‹⟦?x ∈ carrier G; ?y ∈ carrier G⟧ ⟹ inv (?x ⊗ ?y) = inv ?y ⊗ inv ?x›*) is_group (*‹Group.group G›*) assms(1) (*‹subgroup H G›*) g(1) (*‹g ∈ carrier G›*) subgroup.mem_carrier (*‹⟦subgroup ?H ?G; ?x ∈ ?H⟧ ⟹ ?x ∈ carrier ?G›*) by fastforce
  also (*calculation: ‹r1 ⊗ inv r2 = h1 ⊗ g ⊗ (inv g ⊗ inv h2)›*) have " ... =  (h1 ⊗ (g ⊗ inv g) ⊗ inv h2)"
    using h1 (*‹(h1::'a) ∈ (H::'a set)› ‹r1 = h1 ⊗ g›*) h2 (*‹h2 ∈ H› ‹r2 = h2 ⊗ g›*) assms(1) (*‹subgroup H G›*) g(1) (*‹g ∈ carrier G›*) inv_closed (*‹?x ∈ carrier G ⟹ inv ?x ∈ carrier G›*) m_closed (*‹⟦?x ∈ carrier G; ?y ∈ carrier G⟧ ⟹ ?x ⊗ ?y ∈ carrier G›*) monoid.m_assoc (*‹⟦Group.monoid ?G; ?x ∈ carrier ?G; ?y ∈ carrier ?G; ?z ∈ carrier ?G⟧ ⟹ ?x ⊗⇘?G⇙ ?y ⊗⇘?G⇙ ?z = ?x ⊗⇘?G⇙ (?y ⊗⇘?G⇙ ?z)›*) monoid_axioms (*‹Group.monoid G›*) subgroup.mem_carrier (*‹⟦subgroup ?H ?G; ?x ∈ ?H⟧ ⟹ ?x ∈ carrier ?G›*) proof (-)
      (*goal: ‹⟦h1 ∈ H; r1 = h1 ⊗ g; h2 ∈ H; r2 = h2 ⊗ g; subgroup H G; g ∈ carrier G; ⋀x. x ∈ carrier G ⟹ inv x ∈ carrier G; ⋀x y. ⟦x ∈ carrier G; y ∈ carrier G⟧ ⟹ x ⊗ y ∈ carrier G; ⋀G x y z. ⟦Group.monoid G; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G⟧ ⟹ x ⊗⇘G⇙ y ⊗⇘G⇙ z = x ⊗⇘G⇙ (y ⊗⇘G⇙ z); Group.monoid G; ⋀H G x. ⟦subgroup H G; x ∈ H⟧ ⟹ x ∈ carrier G⟧ ⟹ h1 ⊗ g ⊗ (inv g ⊗ inv h2) = h1 ⊗ (g ⊗ inv g) ⊗ inv h2›*)

Types of annotations:

• proof goal
• facts
• calculation mechanism
• ...

Minilang's pipeline is used to transform composite tactics to minimal units and annotate the transitions of proof states per unit.

Each statement has a 30% probability of being annotated with types, and a further 40% probability of being annotated with the sorts of the type variables.

Please cite me!

@misc{xu2025minimalistprooflanguageneural,
      title={A Minimalist Proof Language for Neural Theorem Proving over Isabelle/HOL}, 
      author={Qiyuan Xu and Renxi Wang and Peixin Wang and Haonan Li and Conrad Watt},
      year={2025},
      eprint={2507.18885},
      archivePrefix={arXiv},
      primaryClass={cs.PL},
      url={https://arxiv.org/abs/2507.18885}, 
}