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--- |
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license: lgpl |
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task_categories: |
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- text-generation |
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language: |
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- en |
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tags: |
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- Isabelle |
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- theorem-proving |
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size_categories: |
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- 100M<n<1B |
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--- |
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Extensively annotated Isabelle source code, suitable for pretraining, about 500M tokens in Qwen3's tokenizer. |
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Data source: Isabelle/HOL + AFP 2025-02-12. Cases from the PISA benchmark are removed. |
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Example: |
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```isabelle |
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lemma (in group) diff_neutralizes: ✐‹contributor ‹Paulo EmÃlio de Vilhena›› |
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assumes "subgroup H G" "R ∈ rcosets H" |
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shows "⋀r1 r2. ⟦ r1 ∈ R; r2 ∈ R ⟧ ⟹ r1 ⊗ (inv r2) ∈ H" |
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proof (-) |
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(*goal: ‹⋀r1 r2. ⟦r1 ∈ R; r2 ∈ R⟧ ⟹ r1 ⊗ inv r2 ∈ H›*) |
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fix r1 and r2 |
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assume r1: "r1 ∈ R" and r2: "r2 ∈ R" (*‹(r1::'a) ∈ (R::'a set)› ‹(r2::'a) ∈ (R::'a set)›*) |
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obtain g where g: "g ∈ carrier G" "R = H #> g" |
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(*goal: ‹(⋀g. ⟦g ∈ carrier G; R = H #> g⟧ ⟹ thesis) ⟹ thesis›*) |
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using assms (*‹subgroup (H::'a::type set) G› ‹R ∈ rcosets H›*) unfolding RCOSETS_def |
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(*goal: ‹(⋀g. ⟦g ∈ carrier G; R = H #> g⟧ ⟹ thesis) ⟹ thesis›*) |
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by blast |
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then obtain h1 and h2 where h1: "h1 ∈ H" "r1 = h1 ⊗ g" and h2: "h2 ∈ H" "r2 = h2 ⊗ g" |
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(*goal: ‹(⋀h1 h2. ⟦h1 ∈ H; r1 = h1 ⊗ g; h2 ∈ H; r2 = h2 ⊗ g⟧ ⟹ thesis) ⟹ thesis›*) |
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using r1 (*‹(r1::'a) ∈ (R::'a set)›*) r2 (*‹r2 ∈ R›*) unfolding r_coset_def |
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(*goal: ‹(⋀h1 h2. ⟦h1 ∈ H; r1 = h1 ⊗ g; h2 ∈ H; r2 = h2 ⊗ g⟧ ⟹ thesis) ⟹ thesis›*) |
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by blast |
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hence "r1 ⊗ (inv r2) = (h1 ⊗ g) ⊗ ((inv g) ⊗ (inv h2))" |
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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 |
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also (*calculation: ‹r1 ⊗ inv r2 = h1 ⊗ g ⊗ (inv g ⊗ inv h2)›*) have " ... = (h1 ⊗ (g ⊗ inv g) ⊗ inv h2)" |
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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 (-) |
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(*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›*) |
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``` |
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Types of annotations: |
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- proof goal |
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- facts |
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- calculation mechanism |
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- ... |
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Minilang's pipeline is used to transform composite tactics to minimal units and annotate the transitions of proof states per unit. |
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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. |
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Please cite me! |
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``` |
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@misc{xu2025minimalistprooflanguageneural, |
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title={A Minimalist Proof Language for Neural Theorem Proving over Isabelle/HOL}, |
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author={Qiyuan Xu and Renxi Wang and Peixin Wang and Haonan Li and Conrad Watt}, |
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year={2025}, |
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eprint={2507.18885}, |
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archivePrefix={arXiv}, |
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primaryClass={cs.PL}, |
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url={https://arxiv.org/abs/2507.18885}, |
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} |
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``` |
