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But the vast scope of formal mathematics means that any strong reasoning result obtained in", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 564, 505, 577 ], "spans": [ { "bbox": [ 105, 564, 505, 577 ], "score": 1.0, "content": "it will be more meaningful than comparable results in games (e.g. finding proofs to mathematical", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 576, 506, 588 ], "spans": [ { "bbox": [ 105, 576, 506, 588 ], "score": 1.0, "content": "conjectures), and could even be applicable to important practical problems (e.g. software verification).", "type": "text" } ], "index": 41 } ], "index": 38 }, { "type": "text", "bbox": [ 108, 592, 503, 614 ], "lines": [ { "bbox": [ 106, 590, 505, 604 ], "spans": [ { "bbox": [ 106, 590, 505, 604 ], "score": 1.0, "content": "However, tackling formal mathematics involves two main challenges that we must address in order to", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 603, 215, 615 ], "spans": [ { "bbox": [ 106, 603, 215, 615 ], "score": 1.0, "content": "continue making progress:", "type": "text" } ], "index": 43 } ], "index": 42.5 }, { "type": "text", "bbox": [ 107, 620, 505, 686 ], "lines": [ { "bbox": [ 105, 619, 505, 632 ], "spans": [ { "bbox": [ 105, 619, 505, 632 ], "score": 1.0, "content": "Infinite action space Not only does formal mathematics have an extremely large search space (like", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 631, 506, 642 ], "spans": [ { "bbox": [ 106, 631, 506, 642 ], "score": 1.0, "content": "Go (Silver et al., 2016) for example), it also has an infinite action space. 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One domain where deep learning has not yet", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 437, 507, 451 ], "spans": [ { "bbox": [ 105, 437, 507, 451 ], "score": 1.0, "content": "enjoyed a comparable success is in tasks that require extensive planning and symbolic reasoning,", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 449, 506, 462 ], "spans": [ { "bbox": [ 105, 449, 506, 462 ], "score": 1.0, "content": "with the exception of two-player games (Silver et al., 2016; 2017; Berner et al., 2019; Vinyals et al.,", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 459, 505, 473 ], "spans": [ { "bbox": [ 105, 459, 505, 473 ], "score": 1.0, "content": "2019). 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But the resulting reasoning abilities achieved are limited due to the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 491, 243, 506 ], "spans": [ { "bbox": [ 105, 491, 243, 506 ], "score": 1.0, "content": "relatively narrow scope of games.", "type": "text" } ], "index": 34 } ], "index": 30, "bbox_fs": [ 104, 405, 507, 506 ] }, { "type": "text", "bbox": [ 107, 509, 505, 587 ], "lines": [ { "bbox": [ 105, 508, 506, 523 ], "spans": [ { "bbox": [ 105, 508, 506, 523 ], "score": 1.0, "content": "As such, theorem proving in interactive proof assistants, or formal mathematics, appears as an", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 521, 506, 533 ], "spans": [ { "bbox": [ 105, 521, 506, 533 ], "score": 1.0, "content": "interesting game-like domain to tackle due to its increased scope. The typical tasks consist of", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 531, 505, 543 ], "spans": [ { "bbox": [ 105, 531, 505, 543 ], "score": 1.0, "content": "generating a machine-checkable proof given a formal statements. Like games, formal mathematics", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 542, 504, 555 ], "spans": [ { "bbox": [ 105, 542, 504, 555 ], "score": 1.0, "content": "has an automated way of determining whether a trajectory (i.e. a proof) is successful (i.e. formally", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 553, 505, 565 ], "spans": [ { "bbox": [ 105, 553, 505, 565 ], "score": 1.0, "content": "correct). But the vast scope of formal mathematics means that any strong reasoning result obtained in", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 564, 505, 577 ], "spans": [ { "bbox": [ 105, 564, 505, 577 ], "score": 1.0, "content": "it will be more meaningful than comparable results in games (e.g. finding proofs to mathematical", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 576, 506, 588 ], "spans": [ { "bbox": [ 105, 576, 506, 588 ], "score": 1.0, "content": "conjectures), and could even be applicable to important practical problems (e.g. software verification).", "type": "text" } ], "index": 41 } ], "index": 38, "bbox_fs": [ 105, 508, 506, 588 ] }, { "type": "text", "bbox": [ 108, 592, 503, 614 ], "lines": [ { "bbox": [ 106, 590, 505, 604 ], "spans": [ { "bbox": [ 106, 590, 505, 604 ], "score": 1.0, "content": "However, tackling formal mathematics involves two main challenges that we must address in order to", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 603, 215, 615 ], "spans": [ { "bbox": [ 106, 603, 215, 615 ], "score": 1.0, "content": "continue making progress:", "type": "text" } ], "index": 43 } ], "index": 42.5, "bbox_fs": [ 106, 590, 505, 615 ] }, { "type": "text", "bbox": [ 107, 620, 505, 686 ], "lines": [ { "bbox": [ 105, 619, 505, 632 ], "spans": [ { "bbox": [ 105, 619, 505, 632 ], "score": 1.0, "content": "Infinite action space Not only does formal mathematics have an extremely large search space (like", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 631, 506, 642 ], "spans": [ { "bbox": [ 106, 631, 506, 642 ], "score": 1.0, "content": "Go (Silver et al., 2016) for example), it also has an infinite action space. At each step of proof search,", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 642, 505, 653 ], "spans": [ { "bbox": [ 106, 642, 505, 653 ], "score": 1.0, "content": "the model must choose not from a well-behaved finite set of actions, but a complex and infinite", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 651, 507, 667 ], "spans": [ { "bbox": [ 105, 651, 507, 667 ], "score": 1.0, "content": "set of tactics, potentially involving exogenous mathematical terms that have to be generated (e.g.,", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 663, 506, 676 ], "spans": [ { "bbox": [ 105, 663, 506, 676 ], "score": 1.0, "content": "generating a mathematical statement to be used as a witness, an object used steps such as “there exists", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 675, 461, 687 ], "spans": [ { "bbox": [ 105, 675, 461, 687 ], "score": 1.0, "content": "an x ...”, or a cut, the introduction and the chaining of a lemma in the middle of a proof).", "type": "text" } ], "index": 49 } ], "index": 46.5, "bbox_fs": [ 105, 619, 507, 687 ] }, { "type": "text", "bbox": [ 108, 692, 505, 714 ], "lines": [ { "bbox": [ 106, 691, 505, 704 ], "spans": [ { "bbox": [ 106, 691, 505, 704 ], "score": 1.0, "content": "No direct self-play setup In formal mathematics, a prover is not playing against an opponent but", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 702, 505, 716 ], "spans": [ { "bbox": [ 106, 702, 505, 716 ], "score": 1.0, "content": "against a set of statements to prove. When faced with a statement that is just too hard, there is no", "type": "text" } ], "index": 51 }, { "bbox": [ 106, 81, 505, 95 ], "spans": [ { "bbox": [ 106, 81, 505, 95 ], "score": 1.0, "content": "obvious reframing of the formal mathematics setup that will let the prover generate intermediary", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 93, 505, 107 ], "spans": [ { "bbox": [ 105, 93, 505, 107 ], "score": 1.0, "content": "easier statements to tackle first. This asymmetry prevents naive application of the symmetric self-play", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 106, 104, 294, 117 ], "spans": [ { "bbox": [ 106, 104, 294, 117 ], "score": 1.0, "content": "algorithms commonly used in 2-player games.", "type": "text", "cross_page": true } ], "index": 2 } ], "index": 50.5, "bbox_fs": [ 106, 691, 505, 716 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 108, 82, 504, 116 ], "lines": [ { "bbox": [ 106, 81, 505, 95 ], "spans": [ { "bbox": [ 106, 81, 505, 95 ], "score": 1.0, "content": "obvious reframing of the formal mathematics setup that will let the prover generate intermediary", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 107 ], "spans": [ { "bbox": [ 105, 93, 505, 107 ], "score": 1.0, "content": "easier statements to tackle first. This asymmetry prevents naive application of the symmetric self-play", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 294, 117 ], "spans": [ { "bbox": [ 106, 104, 294, 117 ], "score": 1.0, "content": "algorithms commonly used in 2-player games.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "text", "bbox": [ 107, 122, 505, 286 ], "lines": [ { "bbox": [ 106, 121, 505, 134 ], "spans": [ { "bbox": [ 106, 121, 505, 134 ], "score": 1.0, "content": "These two differences make a naive application of reinforcement learning to formal mathematics", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 131, 505, 145 ], "spans": [ { "bbox": [ 105, 131, 505, 145 ], "score": 1.0, "content": "leave a large room for improvement (Whalen, 2016; Winands et al., 2008). Past work proposed to", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 143, 506, 156 ], "spans": [ { "bbox": [ 105, 143, 506, 156 ], "score": 1.0, "content": "address the infinite action space problem by sampling from a language model (Polu & Sutskever,", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 154, 506, 167 ], "spans": [ { "bbox": [ 106, 154, 506, 167 ], "score": 1.0, "content": "2020), while training such language model requires a large dataset of statements with proof. This", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 164, 506, 179 ], "spans": [ { "bbox": [ 105, 164, 506, 179 ], "score": 1.0, "content": "paper focuses on this second problem and our basis for addressing it is the observation that the key", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 175, 505, 190 ], "spans": [ { "bbox": [ 105, 175, 505, 190 ], "score": 1.0, "content": "role of self-play is to provide an unsupervised curriculum. We propose instead to supply auxiliary", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 506, 201 ], "spans": [ { "bbox": [ 105, 187, 506, 201 ], "score": 1.0, "content": "sets of problem statements (without requiring proofs) of varying difficulty. We empirically show that,", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 198, 505, 211 ], "spans": [ { "bbox": [ 105, 198, 505, 211 ], "score": 1.0, "content": "when the difficulty of these auxiliary problems is varied enough, a simple expert iteration procedure", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 208, 506, 223 ], "spans": [ { "bbox": [ 104, 208, 506, 223 ], "score": 1.0, "content": "is able to solve a curriculum of increasingly difficult problems, eventually generalizing to our target", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 219, 506, 233 ], "spans": [ { "bbox": [ 105, 219, 506, 233 ], "score": 1.0, "content": "distribution. We show that this works with both automatically-generated and manually-curated", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 231, 505, 243 ], "spans": [ { "bbox": [ 106, 231, 505, 243 ], "score": 1.0, "content": "auxiliary distributions of problems and leverage this to achieve state-of-the-art on the miniF2F", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 242, 505, 255 ], "spans": [ { "bbox": [ 106, 242, 505, 255 ], "score": 1.0, "content": "benchmark. Our results suggest that continuous self-improvement in formal mathematics can", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 253, 505, 265 ], "spans": [ { "bbox": [ 105, 253, 505, 265 ], "score": 1.0, "content": "potentially be reduced to the problem of generating such sets of formal statements, which we have", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 263, 506, 276 ], "spans": [ { "bbox": [ 105, 263, 506, 276 ], "score": 1.0, "content": "done in part manually in this work, but could eventually be scaled in the future with more automation", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 275, 506, 287 ], "spans": [ { "bbox": [ 105, 275, 506, 287 ], "score": 1.0, "content": "(such as more domain-specific statements generator or even informal to formal machine translation).", "type": "text" } ], "index": 17 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 291, 505, 347 ], "lines": [ { "bbox": [ 105, 291, 505, 304 ], "spans": [ { "bbox": [ 105, 291, 331, 304 ], "score": 1.0, "content": "miniF2F benchmark In this work, we target the miniF", "type": "text" }, { "bbox": [ 331, 292, 344, 302 ], "score": 0.34, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 345, 291, 505, 304 ], "score": 1.0, "content": "(Zheng et al., 2022) benchmark, which", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 303, 505, 315 ], "spans": [ { "bbox": [ 106, 303, 505, 315 ], "score": 1.0, "content": "consists of 244 validation and 244 test formalized statements of mathematical problems from various", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 313, 505, 327 ], "spans": [ { "bbox": [ 105, 313, 505, 327 ], "score": 1.0, "content": "competitions. We believe it to be a better measure of mathematical reasoning compared to a formal", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 324, 506, 338 ], "spans": [ { "bbox": [ 105, 324, 506, 338 ], "score": 1.0, "content": "library-derived split. Also, the extreme scarcity in formal libraries of this type of problems makes it", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 335, 406, 349 ], "spans": [ { "bbox": [ 105, 335, 406, 349 ], "score": 1.0, "content": "an ideal test-bed for the expert iteration methodology studied in this paper.", "type": "text" } ], "index": 22 } ], "index": 20 }, { "type": "title", "bbox": [ 108, 366, 210, 379 ], "lines": [ { "bbox": [ 104, 365, 213, 382 ], "spans": [ { "bbox": [ 104, 365, 213, 382 ], "score": 1.0, "content": "2 RELATED WORK", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 394, 505, 449 ], "lines": [ { "bbox": [ 106, 394, 505, 406 ], "spans": [ { "bbox": [ 106, 394, 505, 406 ], "score": 1.0, "content": "Our work strongly relies on, and can be seen as a natural continuation of the work presented in the", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 405, 505, 418 ], "spans": [ { "bbox": [ 105, 405, 505, 418 ], "score": 1.0, "content": "original GPT-f paper (Polu & Sutskever, 2020) which studies the use of language models to generate", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 416, 505, 428 ], "spans": [ { "bbox": [ 106, 416, 505, 428 ], "score": 1.0, "content": "tactics, the PACT paper (Han et al., 2022) which applies GPT-f to Lean and studies the benefits from", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 427, 505, 440 ], "spans": [ { "bbox": [ 105, 427, 505, 440 ], "score": 1.0, "content": "co-training on self-supervised objectives, and the miniF2F benchmark (Zheng et al., 2022). We", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 438, 297, 451 ], "spans": [ { "bbox": [ 105, 438, 297, 451 ], "score": 1.0, "content": "present additional related work in Appendix A.", "type": "text" } ], "index": 28 } ], "index": 26 }, { "type": "title", "bbox": [ 108, 469, 251, 482 ], "lines": [ { "bbox": [ 105, 468, 253, 484 ], "spans": [ { "bbox": [ 105, 468, 253, 484 ], "score": 1.0, "content": "3 FORMAL ENVIRONMENT", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 496, 505, 595 ], "lines": [ { "bbox": [ 105, 495, 505, 509 ], "spans": [ { "bbox": [ 105, 495, 505, 509 ], "score": 1.0, "content": "We choose Lean (de Moura et al., 2015; lea) as our formal environment. Unlike Metamath (Megill", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 507, 506, 520 ], "spans": [ { "bbox": [ 105, 507, 506, 520 ], "score": 1.0, "content": "& Wheeler, 2019) , which has been studied in the original GPT-f paper (Polu & Sutskever, 2020),", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 519, 505, 530 ], "spans": [ { "bbox": [ 106, 519, 505, 530 ], "score": 1.0, "content": "Lean benefits from high-level tactics which were shown to be beneficial in the context of the miniF2F", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 528, 507, 543 ], "spans": [ { "bbox": [ 105, 528, 507, 543 ], "score": 1.0, "content": "benchmark. Also, Lean has recently received a lot of attention from the mathematical community,", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 541, 506, 553 ], "spans": [ { "bbox": [ 106, 541, 506, 553 ], "score": 1.0, "content": "thanks to projects such as the Perfectoid Spaces (Buzzard et al., 2019) and the Liquid Tensor", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 550, 506, 565 ], "spans": [ { "bbox": [ 105, 550, 506, 565 ], "score": 1.0, "content": "experiment (Scholze, 2020), and benefits from a vibrant community of hundreds of contributors to its", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 561, 505, 575 ], "spans": [ { "bbox": [ 105, 561, 505, 575 ], "score": 1.0, "content": "main mathematical library called mathlib. We refer to the PACT paper’s Background section (Han", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 572, 505, 587 ], "spans": [ { "bbox": [ 105, 572, 505, 587 ], "score": 1.0, "content": "et al., 2022) for a detailed introduction to Lean in the context of neural theorem proving. We refer to", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 584, 394, 598 ], "spans": [ { "bbox": [ 105, 584, 394, 598 ], "score": 1.0, "content": "Appendix D for an illustration of miniF2F input and Lean environment.", "type": "text" } ], "index": 38 } ], "index": 34 }, { "type": "text", "bbox": [ 107, 601, 504, 667 ], "lines": [ { "bbox": [ 105, 599, 506, 615 ], "spans": [ { "bbox": [ 105, 599, 506, 615 ], "score": 1.0, "content": "lean-gym In the PACT paper (Han et al., 2022), proof search is performed by the Lean runtime using", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 611, 506, 625 ], "spans": [ { "bbox": [ 105, 611, 506, 625 ], "score": 1.0, "content": "the LEANSTEP environment, with a generic backend interface to models. While easy to use–one", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 623, 506, 636 ], "spans": [ { "bbox": [ 105, 623, 506, 636 ], "score": 1.0, "content": "just needs to plug in their model–this approach makes it difficult to alter and iterate on the search", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 635, 506, 646 ], "spans": [ { "bbox": [ 105, 635, 506, 646 ], "score": 1.0, "content": "procedure because it is programmed in Lean (which is not designed or intended for cluster-wide", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 645, 506, 658 ], "spans": [ { "bbox": [ 105, 645, 506, 658 ], "score": 1.0, "content": "parallelised I/O intensive tasks), and the coupling of the search procedure with the Lean runtime", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 656, 401, 667 ], "spans": [ { "bbox": [ 106, 656, 401, 667 ], "score": 1.0, "content": "introduces challenges when scaling to a large number of parallel workers.", "type": "text" } ], "index": 44 } ], "index": 41.5 }, { "type": "text", "bbox": [ 108, 672, 505, 706 ], "lines": [ { "bbox": [ 106, 673, 505, 685 ], "spans": [ { "bbox": [ 106, 673, 505, 685 ], "score": 1.0, "content": "To solve these issues we implemented lean-gym1 – a simple REPL interface over the standard", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 684, 505, 696 ], "spans": [ { "bbox": [ 106, 684, 505, 696 ], "score": 1.0, "content": "input/output implemented in Lean directly. We present lean-gym’s API and discuss some of its", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 695, 279, 707 ], "spans": [ { "bbox": [ 106, 695, 279, 707 ], "score": 1.0, "content": "advantages and limitations in Appendix B.", "type": "text" } ], "index": 47 } ], "index": 46 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 120, 721, 277, 732 ], "lines": [ { "bbox": [ 119, 720, 278, 734 ], "spans": [ { "bbox": [ 119, 720, 278, 734 ], "score": 1.0, "content": "1https://github.com/openai/lean-gym", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 763 ], "spans": [ { "bbox": [ 301, 750, 310, 763 ], "score": 1.0, "content": "2", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 108, 82, 504, 116 ], "lines": [], "index": 1, "bbox_fs": [ 105, 81, 505, 117 ], "lines_deleted": true }, { "type": "text", "bbox": [ 107, 122, 505, 286 ], "lines": [ { "bbox": [ 106, 121, 505, 134 ], "spans": [ { "bbox": [ 106, 121, 505, 134 ], "score": 1.0, "content": "These two differences make a naive application of reinforcement learning to formal mathematics", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 131, 505, 145 ], "spans": [ { "bbox": [ 105, 131, 505, 145 ], "score": 1.0, "content": "leave a large room for improvement (Whalen, 2016; Winands et al., 2008). Past work proposed to", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 143, 506, 156 ], "spans": [ { "bbox": [ 105, 143, 506, 156 ], "score": 1.0, "content": "address the infinite action space problem by sampling from a language model (Polu & Sutskever,", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 154, 506, 167 ], "spans": [ { "bbox": [ 106, 154, 506, 167 ], "score": 1.0, "content": "2020), while training such language model requires a large dataset of statements with proof. This", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 164, 506, 179 ], "spans": [ { "bbox": [ 105, 164, 506, 179 ], "score": 1.0, "content": "paper focuses on this second problem and our basis for addressing it is the observation that the key", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 175, 505, 190 ], "spans": [ { "bbox": [ 105, 175, 505, 190 ], "score": 1.0, "content": "role of self-play is to provide an unsupervised curriculum. We propose instead to supply auxiliary", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 506, 201 ], "spans": [ { "bbox": [ 105, 187, 506, 201 ], "score": 1.0, "content": "sets of problem statements (without requiring proofs) of varying difficulty. We empirically show that,", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 198, 505, 211 ], "spans": [ { "bbox": [ 105, 198, 505, 211 ], "score": 1.0, "content": "when the difficulty of these auxiliary problems is varied enough, a simple expert iteration procedure", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 208, 506, 223 ], "spans": [ { "bbox": [ 104, 208, 506, 223 ], "score": 1.0, "content": "is able to solve a curriculum of increasingly difficult problems, eventually generalizing to our target", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 219, 506, 233 ], "spans": [ { "bbox": [ 105, 219, 506, 233 ], "score": 1.0, "content": "distribution. We show that this works with both automatically-generated and manually-curated", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 231, 505, 243 ], "spans": [ { "bbox": [ 106, 231, 505, 243 ], "score": 1.0, "content": "auxiliary distributions of problems and leverage this to achieve state-of-the-art on the miniF2F", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 242, 505, 255 ], "spans": [ { "bbox": [ 106, 242, 505, 255 ], "score": 1.0, "content": "benchmark. Our results suggest that continuous self-improvement in formal mathematics can", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 253, 505, 265 ], "spans": [ { "bbox": [ 105, 253, 505, 265 ], "score": 1.0, "content": "potentially be reduced to the problem of generating such sets of formal statements, which we have", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 263, 506, 276 ], "spans": [ { "bbox": [ 105, 263, 506, 276 ], "score": 1.0, "content": "done in part manually in this work, but could eventually be scaled in the future with more automation", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 275, 506, 287 ], "spans": [ { "bbox": [ 105, 275, 506, 287 ], "score": 1.0, "content": "(such as more domain-specific statements generator or even informal to formal machine translation).", "type": "text" } ], "index": 17 } ], "index": 10, "bbox_fs": [ 104, 121, 506, 287 ] }, { "type": "text", "bbox": [ 107, 291, 505, 347 ], "lines": [ { "bbox": [ 105, 291, 505, 304 ], "spans": [ { "bbox": [ 105, 291, 331, 304 ], "score": 1.0, "content": "miniF2F benchmark In this work, we target the miniF", "type": "text" }, { "bbox": [ 331, 292, 344, 302 ], "score": 0.34, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 345, 291, 505, 304 ], "score": 1.0, "content": "(Zheng et al., 2022) benchmark, which", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 303, 505, 315 ], "spans": [ { "bbox": [ 106, 303, 505, 315 ], "score": 1.0, "content": "consists of 244 validation and 244 test formalized statements of mathematical problems from various", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 313, 505, 327 ], "spans": [ { "bbox": [ 105, 313, 505, 327 ], "score": 1.0, "content": "competitions. We believe it to be a better measure of mathematical reasoning compared to a formal", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 324, 506, 338 ], "spans": [ { "bbox": [ 105, 324, 506, 338 ], "score": 1.0, "content": "library-derived split. Also, the extreme scarcity in formal libraries of this type of problems makes it", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 335, 406, 349 ], "spans": [ { "bbox": [ 105, 335, 406, 349 ], "score": 1.0, "content": "an ideal test-bed for the expert iteration methodology studied in this paper.", "type": "text" } ], "index": 22 } ], "index": 20, "bbox_fs": [ 105, 291, 506, 349 ] }, { "type": "title", "bbox": [ 108, 366, 210, 379 ], "lines": [ { "bbox": [ 104, 365, 213, 382 ], "spans": [ { "bbox": [ 104, 365, 213, 382 ], "score": 1.0, "content": "2 RELATED WORK", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 394, 505, 449 ], "lines": [ { "bbox": [ 106, 394, 505, 406 ], "spans": [ { "bbox": [ 106, 394, 505, 406 ], "score": 1.0, "content": "Our work strongly relies on, and can be seen as a natural continuation of the work presented in the", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 405, 505, 418 ], "spans": [ { "bbox": [ 105, 405, 505, 418 ], "score": 1.0, "content": "original GPT-f paper (Polu & Sutskever, 2020) which studies the use of language models to generate", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 416, 505, 428 ], "spans": [ { "bbox": [ 106, 416, 505, 428 ], "score": 1.0, "content": "tactics, the PACT paper (Han et al., 2022) which applies GPT-f to Lean and studies the benefits from", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 427, 505, 440 ], "spans": [ { "bbox": [ 105, 427, 505, 440 ], "score": 1.0, "content": "co-training on self-supervised objectives, and the miniF2F benchmark (Zheng et al., 2022). We", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 438, 297, 451 ], "spans": [ { "bbox": [ 105, 438, 297, 451 ], "score": 1.0, "content": "present additional related work in Appendix A.", "type": "text" } ], "index": 28 } ], "index": 26, "bbox_fs": [ 105, 394, 505, 451 ] }, { "type": "title", "bbox": [ 108, 469, 251, 482 ], "lines": [ { "bbox": [ 105, 468, 253, 484 ], "spans": [ { "bbox": [ 105, 468, 253, 484 ], "score": 1.0, "content": "3 FORMAL ENVIRONMENT", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 496, 505, 595 ], "lines": [ { "bbox": [ 105, 495, 505, 509 ], "spans": [ { "bbox": [ 105, 495, 505, 509 ], "score": 1.0, "content": "We choose Lean (de Moura et al., 2015; lea) as our formal environment. Unlike Metamath (Megill", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 507, 506, 520 ], "spans": [ { "bbox": [ 105, 507, 506, 520 ], "score": 1.0, "content": "& Wheeler, 2019) , which has been studied in the original GPT-f paper (Polu & Sutskever, 2020),", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 519, 505, 530 ], "spans": [ { "bbox": [ 106, 519, 505, 530 ], "score": 1.0, "content": "Lean benefits from high-level tactics which were shown to be beneficial in the context of the miniF2F", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 528, 507, 543 ], "spans": [ { "bbox": [ 105, 528, 507, 543 ], "score": 1.0, "content": "benchmark. Also, Lean has recently received a lot of attention from the mathematical community,", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 541, 506, 553 ], "spans": [ { "bbox": [ 106, 541, 506, 553 ], "score": 1.0, "content": "thanks to projects such as the Perfectoid Spaces (Buzzard et al., 2019) and the Liquid Tensor", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 550, 506, 565 ], "spans": [ { "bbox": [ 105, 550, 506, 565 ], "score": 1.0, "content": "experiment (Scholze, 2020), and benefits from a vibrant community of hundreds of contributors to its", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 561, 505, 575 ], "spans": [ { "bbox": [ 105, 561, 505, 575 ], "score": 1.0, "content": "main mathematical library called mathlib. We refer to the PACT paper’s Background section (Han", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 572, 505, 587 ], "spans": [ { "bbox": [ 105, 572, 505, 587 ], "score": 1.0, "content": "et al., 2022) for a detailed introduction to Lean in the context of neural theorem proving. We refer to", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 584, 394, 598 ], "spans": [ { "bbox": [ 105, 584, 394, 598 ], "score": 1.0, "content": "Appendix D for an illustration of miniF2F input and Lean environment.", "type": "text" } ], "index": 38 } ], "index": 34, "bbox_fs": [ 105, 495, 507, 598 ] }, { "type": "text", "bbox": [ 107, 601, 504, 667 ], "lines": [ { "bbox": [ 105, 599, 506, 615 ], "spans": [ { "bbox": [ 105, 599, 506, 615 ], "score": 1.0, "content": "lean-gym In the PACT paper (Han et al., 2022), proof search is performed by the Lean runtime using", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 611, 506, 625 ], "spans": [ { "bbox": [ 105, 611, 506, 625 ], "score": 1.0, "content": "the LEANSTEP environment, with a generic backend interface to models. While easy to use–one", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 623, 506, 636 ], "spans": [ { "bbox": [ 105, 623, 506, 636 ], "score": 1.0, "content": "just needs to plug in their model–this approach makes it difficult to alter and iterate on the search", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 635, 506, 646 ], "spans": [ { "bbox": [ 105, 635, 506, 646 ], "score": 1.0, "content": "procedure because it is programmed in Lean (which is not designed or intended for cluster-wide", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 645, 506, 658 ], "spans": [ { "bbox": [ 105, 645, 506, 658 ], "score": 1.0, "content": "parallelised I/O intensive tasks), and the coupling of the search procedure with the Lean runtime", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 656, 401, 667 ], "spans": [ { "bbox": [ 106, 656, 401, 667 ], "score": 1.0, "content": "introduces challenges when scaling to a large number of parallel workers.", "type": "text" } ], "index": 44 } ], "index": 41.5, "bbox_fs": [ 105, 599, 506, 667 ] }, { "type": "text", "bbox": [ 108, 672, 505, 706 ], "lines": [ { "bbox": [ 106, 673, 505, 685 ], "spans": [ { "bbox": [ 106, 673, 505, 685 ], "score": 1.0, "content": "To solve these issues we implemented lean-gym1 – a simple REPL interface over the standard", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 684, 505, 696 ], "spans": [ { "bbox": [ 106, 684, 505, 696 ], "score": 1.0, "content": "input/output implemented in Lean directly. We present lean-gym’s API and discuss some of its", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 695, 279, 707 ], "spans": [ { "bbox": [ 106, 695, 279, 707 ], "score": 1.0, "content": "advantages and limitations in Appendix B.", "type": "text" } ], "index": 47 } ], "index": 46, "bbox_fs": [ 106, 673, 505, 707 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 160 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "Proof extraction We rely on the proof extraction methodology presented in the PACT paper (Han", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 94, 505, 106 ], "spans": [ { "bbox": [ 105, 94, 505, 106 ], "score": 1.0, "content": "et al., 2022) to extract human tactic proof steps from mathlib (the tactic dataset) as well as the", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 219, 117 ], "score": 1.0, "content": "various other proof artifacts", "type": "text" }, { "bbox": [ 219, 105, 241, 115 ], "score": 0.38, "content": "( \\mathsf { m i x } 1", "type": "inline_equation" }, { "bbox": [ 241, 104, 258, 117 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 259, 105, 281, 115 ], "score": 0.51, "content": "\\mathfrak { m i x 2 }", "type": "inline_equation" }, { "bbox": [ 281, 104, 505, 117 ], "score": 1.0, "content": "datasets). We also extract mathlib-{train, valid, test}, the", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 116, 505, 127 ], "spans": [ { "bbox": [ 106, 116, 505, 127 ], "score": 1.0, "content": "set of statements from mathlib along the split proposed in Han et al. (2022) (the validation and test", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 127, 505, 138 ], "spans": [ { "bbox": [ 106, 127, 505, 138 ], "score": 1.0, "content": "splits of tactic, mix1, mix2 being aligned with mathlib-{valid, test} as the splits are determined", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 137, 505, 150 ], "spans": [ { "bbox": [ 105, 137, 505, 150 ], "score": 1.0, "content": "by declaration name hashes (across all data sources including proof-term mining) as opposed to", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 148, 256, 161 ], "spans": [ { "bbox": [ 105, 148, 256, 161 ], "score": 1.0, "content": "individual proof steps or data-points.", "type": "text" } ], "index": 6 } ], "index": 3 }, { "type": "title", "bbox": [ 108, 176, 226, 189 ], "lines": [ { "bbox": [ 105, 175, 227, 190 ], "spans": [ { "bbox": [ 105, 175, 227, 190 ], "score": 1.0, "content": "4 EXPERT ITERATION", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 201, 505, 268 ], "lines": [ { "bbox": [ 105, 200, 506, 215 ], "spans": [ { "bbox": [ 105, 200, 506, 215 ], "score": 1.0, "content": "Expert iteration was introduced in Silver et al. (2017) and broadly consists in iteratively training", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 212, 505, 225 ], "spans": [ { "bbox": [ 105, 212, 505, 225 ], "score": 1.0, "content": "models on their previously sampled trajectories, to achieve continuous improvement. In this section", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 224, 505, 237 ], "spans": [ { "bbox": [ 105, 224, 505, 237 ], "score": 1.0, "content": "we present our expert iteration methodology, including the models and pre-training strategies. We use", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 234, 505, 247 ], "spans": [ { "bbox": [ 105, 234, 505, 247 ], "score": 1.0, "content": "decoder-only Transformers similar to GPT-3 (Brown et al., 2020). Throughout this paper we focus", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 244, 505, 257 ], "spans": [ { "bbox": [ 105, 244, 505, 257 ], "score": 1.0, "content": "on a model with 36 layers and 774 million trainable parameters (referred to as the 700m model in the", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 257, 267, 268 ], "spans": [ { "bbox": [ 106, 257, 267, 268 ], "score": 1.0, "content": "GPT-f paper (Polu & Sutskever, 2020)).", "type": "text" } ], "index": 13 } ], "index": 10.5 }, { "type": "title", "bbox": [ 107, 282, 197, 293 ], "lines": [ { "bbox": [ 106, 281, 197, 294 ], "spans": [ { "bbox": [ 106, 281, 197, 294 ], "score": 1.0, "content": "4.1 PRE-TRAINING", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 107, 302, 505, 336 ], "lines": [ { "bbox": [ 106, 302, 505, 315 ], "spans": [ { "bbox": [ 106, 302, 505, 315 ], "score": 1.0, "content": "We pre-train our models successively on GPT-3’s post-processed version of CommonCrawl (for 300B", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 313, 505, 326 ], "spans": [ { "bbox": [ 106, 313, 505, 326 ], "score": 1.0, "content": "tokens) and an updated version of WebMath (Polu & Sutskever, 2020) (for 72B tokens) whose mix is", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 325, 211, 336 ], "spans": [ { "bbox": [ 105, 325, 211, 336 ], "score": 1.0, "content": "presented in Appendix C.", "type": "text" } ], "index": 17 } ], "index": 16 }, { "type": "title", "bbox": [ 108, 350, 230, 361 ], "lines": [ { "bbox": [ 106, 349, 231, 362 ], "spans": [ { "bbox": [ 106, 349, 231, 362 ], "score": 1.0, "content": "4.2 TRAINING OBJECTIVES", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 370, 505, 414 ], "lines": [ { "bbox": [ 105, 370, 506, 383 ], "spans": [ { "bbox": [ 105, 370, 506, 383 ], "score": 1.0, "content": "Proofstep objective The proofstep objective, introduced in Polu & Sutskever (2020), consists in", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 382, 506, 394 ], "spans": [ { "bbox": [ 105, 382, 506, 394 ], "score": 1.0, "content": "generating a PROOFSTEP (a Lean tactic) given a GOAL (a Lean tactic state). We also condition this", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 393, 506, 406 ], "spans": [ { "bbox": [ 105, 393, 506, 406 ], "score": 1.0, "content": "objective on the current DECLARATION (a Lean theorem name), which remains the same throughout a", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 404, 424, 415 ], "spans": [ { "bbox": [ 105, 404, 424, 415 ], "score": 1.0, "content": "proof search: DECL GOAL PROOFSTEP .", "type": "text" } ], "index": 22 } ], "index": 20.5 }, { "type": "text", "bbox": [ 107, 420, 505, 520 ], "lines": [ { "bbox": [ 106, 420, 505, 432 ], "spans": [ { "bbox": [ 106, 420, 505, 432 ], "score": 1.0, "content": "The rationale for conditioning on the declaration name is to hint our models on the position of the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 430, 505, 444 ], "spans": [ { "bbox": [ 105, 430, 505, 444 ], "score": 1.0, "content": "current declaration in the mathlib library. It can be considered as a weak proxy signal for the large", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 442, 505, 454 ], "spans": [ { "bbox": [ 105, 442, 505, 454 ], "score": 1.0, "content": "amount of information not shown to the model (the full environment consisting of the available", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 453, 505, 465 ], "spans": [ { "bbox": [ 106, 453, 505, 465 ], "score": 1.0, "content": "imports and currently open declarations such as module names, notations, declared instances, ...). The", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 463, 507, 477 ], "spans": [ { "bbox": [ 105, 463, 507, 477 ], "score": 1.0, "content": "declaration name lets models at least in principle memorize and then retrieve some of that information,", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 476, 505, 487 ], "spans": [ { "bbox": [ 106, 476, 505, 487 ], "score": 1.0, "content": "knowing that lean-gym errors if a theorem or definition that is not available in the environment", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 487, 505, 498 ], "spans": [ { "bbox": [ 106, 487, 505, 498 ], "score": 1.0, "content": "associated with the current declaration is used by tactics generated by our models. Also note that", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 497, 505, 510 ], "spans": [ { "bbox": [ 105, 497, 367, 510 ], "score": 1.0, "content": "conversely to Polu & Sutskever (2020) and like Han et al. (2022)", "type": "text" }, { "bbox": [ 367, 498, 399, 508 ], "score": 0.49, "content": "{ < } G O A L >", "type": "inline_equation" }, { "bbox": [ 399, 497, 505, 510 ], "score": 1.0, "content": "is not necessarily a single", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 508, 385, 521 ], "spans": [ { "bbox": [ 105, 508, 385, 521 ], "score": 1.0, "content": "goal but a Lean tactic state, which possibly comprises multiple goals.", "type": "text" } ], "index": 31 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 525, 505, 568 ], "lines": [ { "bbox": [ 105, 524, 505, 538 ], "spans": [ { "bbox": [ 105, 524, 505, 538 ], "score": 1.0, "content": "Proofsize objective We depart from Polu & Sutskever (2020) and use a proofsize objective to", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 536, 505, 549 ], "spans": [ { "bbox": [ 105, 536, 505, 549 ], "score": 1.0, "content": "guide our proof searches, which consists in generating one token that represents a proof size", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 547, 505, 559 ], "spans": [ { "bbox": [ 105, 547, 473, 559 ], "score": 1.0, "content": "estimate bucket for the current goal (Lean tactic state): DECL GOAL", "type": "text" }, { "bbox": [ 473, 547, 505, 558 ], "score": 0.38, "content": "{ < } G O A L >", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 106, 559, 277, 569 ], "spans": [ { "bbox": [ 106, 559, 277, 569 ], "score": 1.0, "content": "PROOFSIZE ", "type": "text" } ], "index": 35 } ], "index": 33.5 }, { "type": "text", "bbox": [ 107, 574, 505, 619 ], "lines": [ { "bbox": [ 105, 574, 505, 587 ], "spans": [ { "bbox": [ 105, 574, 172, 587 ], "score": 1.0, "content": "For a given goal", "type": "text" }, { "bbox": [ 172, 577, 179, 586 ], "score": 0.71, "content": "g", "type": "inline_equation" }, { "bbox": [ 179, 574, 505, 587 ], "score": 1.0, "content": ", either the goal was proved as part of the proof search and we denote its proof size", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 586, 506, 599 ], "spans": [ { "bbox": [ 105, 586, 432, 599 ], "score": 1.0, "content": "(the number of tactic applications (compounded Lean tactics counting as one)) as", "type": "text" }, { "bbox": [ 433, 586, 456, 598 ], "score": 0.91, "content": "p s ( g )", "type": "inline_equation" }, { "bbox": [ 456, 586, 506, 599 ], "score": 1.0, "content": ", or the goal", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 597, 506, 609 ], "spans": [ { "bbox": [ 105, 597, 506, 609 ], "score": 1.0, "content": "was not proved in which case we assign the goal to a bucket that virtually represents \"infinite\" proof", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 608, 132, 620 ], "spans": [ { "bbox": [ 105, 608, 132, 620 ], "score": 1.0, "content": "sizes.", "type": "text" } ], "index": 39 } ], "index": 37.5 }, { "type": "text", "bbox": [ 107, 624, 505, 669 ], "lines": [ { "bbox": [ 105, 623, 505, 638 ], "spans": [ { "bbox": [ 105, 623, 186, 638 ], "score": 1.0, "content": "We use 11 buckets", "type": "text" }, { "bbox": [ 187, 625, 235, 635 ], "score": 0.88, "content": "B = 0 . . . 1 0", "type": "inline_equation" }, { "bbox": [ 236, 623, 379, 638 ], "score": 1.0, "content": "and compute the proofsize bucket", "type": "text" }, { "bbox": [ 379, 624, 397, 636 ], "score": 0.9, "content": "b ( g )", "type": "inline_equation" }, { "bbox": [ 397, 623, 442, 638 ], "score": 1.0, "content": "for a goal", "type": "text" }, { "bbox": [ 442, 627, 449, 636 ], "score": 0.76, "content": "g", "type": "inline_equation" }, { "bbox": [ 449, 623, 505, 638 ], "score": 1.0, "content": "by assigning", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 635, 505, 648 ], "spans": [ { "bbox": [ 105, 635, 505, 648 ], "score": 1.0, "content": "infinite proof sizes to bucket 0, all proof sizes over 20 to bucket 1 and linearly projecting proof sizes", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 646, 506, 659 ], "spans": [ { "bbox": [ 105, 646, 264, 659 ], "score": 1.0, "content": "lower than 20 on the remaining buckets", "type": "text" }, { "bbox": [ 264, 647, 298, 658 ], "score": 0.69, "content": "2 , . . . , 1 0", "type": "inline_equation" }, { "bbox": [ 298, 646, 506, 659 ], "score": 1.0, "content": "(10 being the bucket for the shortest proof sizes). In", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 658, 460, 669 ], "spans": [ { "bbox": [ 105, 658, 356, 669 ], "score": 1.0, "content": "practice, when training and sampling from the model, we map", "type": "text" }, { "bbox": [ 357, 658, 366, 667 ], "score": 0.8, "content": "B", "type": "inline_equation" }, { "bbox": [ 366, 658, 421, 669 ], "score": 1.0, "content": "to the tokens", "type": "text" }, { "bbox": [ 421, 658, 456, 668 ], "score": 0.72, "content": "\\therefore \\mathrm { A \\Omega } . . . \\mathrm { K \\Omega }", "type": "inline_equation" }, { "bbox": [ 456, 658, 460, 669 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 43 } ], "index": 41.5 }, { "type": "text", "bbox": [ 107, 674, 505, 732 ], "lines": [ { "bbox": [ 105, 674, 506, 687 ], "spans": [ { "bbox": [ 105, 674, 506, 687 ], "score": 1.0, "content": "To value goals as we run proof searches, we sample the proofsize bucket token and record the", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 684, 505, 699 ], "spans": [ { "bbox": [ 106, 684, 153, 699 ], "score": 1.0, "content": "probability", "type": "text" }, { "bbox": [ 153, 685, 176, 696 ], "score": 0.89, "content": "p _ { b } ( g )", "type": "inline_equation" }, { "bbox": [ 176, 684, 505, 699 ], "score": 1.0, "content": "for each viable bucket and use them to get a weighted average with the following", "type": "text" } ], "index": 45 }, { "bbox": [ 104, 694, 507, 713 ], "spans": [ { "bbox": [ 104, 694, 144, 713 ], "score": 1.0, "content": "formula:", "type": "text" }, { "bbox": [ 144, 696, 254, 711 ], "score": 0.92, "content": "\\begin{array} { r } { \\dot { v } ( \\dot { g } ) = \\frac { 1 } { \\# B } \\sum _ { b \\in B } p _ { b } ( g ) \\cdot b } \\end{array}", "type": "inline_equation" }, { "bbox": [ 255, 694, 403, 713 ], "score": 1.0, "content": ". As an example, if the model assigns", "type": "text" }, { "bbox": [ 403, 697, 433, 708 ], "score": 0.93, "content": "p _ { 0 } = 1", "type": "inline_equation" }, { "bbox": [ 433, 694, 462, 713 ], "score": 1.0, "content": "(hence", "type": "text" }, { "bbox": [ 462, 697, 501, 709 ], "score": 0.9, "content": "p _ { b \\neq 0 } = 0", "type": "inline_equation" }, { "bbox": [ 502, 694, 507, 713 ], "score": 1.0, "content": ")", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 708, 506, 723 ], "spans": [ { "bbox": [ 105, 708, 126, 723 ], "score": 1.0, "content": "then", "type": "text" }, { "bbox": [ 127, 709, 164, 721 ], "score": 0.91, "content": "v ( g ) = 0", "type": "inline_equation" }, { "bbox": [ 164, 708, 299, 723 ], "score": 1.0, "content": ". Conversely if the model assigns", "type": "text" }, { "bbox": [ 299, 710, 332, 721 ], "score": 0.9, "content": "p _ { 1 0 } = 1", "type": "inline_equation" }, { "bbox": [ 333, 708, 506, 723 ], "score": 1.0, "content": "(10 being the bucket for the shortest proof", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 719, 193, 733 ], "spans": [ { "bbox": [ 105, 719, 151, 733 ], "score": 1.0, "content": "sizes) then", "type": "text" }, { "bbox": [ 151, 721, 189, 732 ], "score": 0.92, "content": "v ( g ) = 1", "type": "inline_equation" }, { "bbox": [ 189, 719, 193, 733 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 48 } ], "index": 46 } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 761 ], "spans": [ { "bbox": [ 302, 750, 309, 761 ], "score": 1.0, "content": "3", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 160 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "Proof extraction We rely on the proof extraction methodology presented in the PACT paper (Han", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 94, 505, 106 ], "spans": [ { "bbox": [ 105, 94, 505, 106 ], "score": 1.0, "content": "et al., 2022) to extract human tactic proof steps from mathlib (the tactic dataset) as well as the", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 219, 117 ], "score": 1.0, "content": "various other proof artifacts", "type": "text" }, { "bbox": [ 219, 105, 241, 115 ], "score": 0.38, "content": "( \\mathsf { m i x } 1", "type": "inline_equation" }, { "bbox": [ 241, 104, 258, 117 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 259, 105, 281, 115 ], "score": 0.51, "content": "\\mathfrak { m i x 2 }", "type": "inline_equation" }, { "bbox": [ 281, 104, 505, 117 ], "score": 1.0, "content": "datasets). We also extract mathlib-{train, valid, test}, the", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 116, 505, 127 ], "spans": [ { "bbox": [ 106, 116, 505, 127 ], "score": 1.0, "content": "set of statements from mathlib along the split proposed in Han et al. (2022) (the validation and test", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 127, 505, 138 ], "spans": [ { "bbox": [ 106, 127, 505, 138 ], "score": 1.0, "content": "splits of tactic, mix1, mix2 being aligned with mathlib-{valid, test} as the splits are determined", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 137, 505, 150 ], "spans": [ { "bbox": [ 105, 137, 505, 150 ], "score": 1.0, "content": "by declaration name hashes (across all data sources including proof-term mining) as opposed to", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 148, 256, 161 ], "spans": [ { "bbox": [ 105, 148, 256, 161 ], "score": 1.0, "content": "individual proof steps or data-points.", "type": "text" } ], "index": 6 } ], "index": 3, "bbox_fs": [ 105, 82, 505, 161 ] }, { "type": "title", "bbox": [ 108, 176, 226, 189 ], "lines": [ { "bbox": [ 105, 175, 227, 190 ], "spans": [ { "bbox": [ 105, 175, 227, 190 ], "score": 1.0, "content": "4 EXPERT ITERATION", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 201, 505, 268 ], "lines": [ { "bbox": [ 105, 200, 506, 215 ], "spans": [ { "bbox": [ 105, 200, 506, 215 ], "score": 1.0, "content": "Expert iteration was introduced in Silver et al. (2017) and broadly consists in iteratively training", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 212, 505, 225 ], "spans": [ { "bbox": [ 105, 212, 505, 225 ], "score": 1.0, "content": "models on their previously sampled trajectories, to achieve continuous improvement. In this section", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 224, 505, 237 ], "spans": [ { "bbox": [ 105, 224, 505, 237 ], "score": 1.0, "content": "we present our expert iteration methodology, including the models and pre-training strategies. We use", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 234, 505, 247 ], "spans": [ { "bbox": [ 105, 234, 505, 247 ], "score": 1.0, "content": "decoder-only Transformers similar to GPT-3 (Brown et al., 2020). Throughout this paper we focus", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 244, 505, 257 ], "spans": [ { "bbox": [ 105, 244, 505, 257 ], "score": 1.0, "content": "on a model with 36 layers and 774 million trainable parameters (referred to as the 700m model in the", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 257, 267, 268 ], "spans": [ { "bbox": [ 106, 257, 267, 268 ], "score": 1.0, "content": "GPT-f paper (Polu & Sutskever, 2020)).", "type": "text" } ], "index": 13 } ], "index": 10.5, "bbox_fs": [ 105, 200, 506, 268 ] }, { "type": "title", "bbox": [ 107, 282, 197, 293 ], "lines": [ { "bbox": [ 106, 281, 197, 294 ], "spans": [ { "bbox": [ 106, 281, 197, 294 ], "score": 1.0, "content": "4.1 PRE-TRAINING", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 107, 302, 505, 336 ], "lines": [ { "bbox": [ 106, 302, 505, 315 ], "spans": [ { "bbox": [ 106, 302, 505, 315 ], "score": 1.0, "content": "We pre-train our models successively on GPT-3’s post-processed version of CommonCrawl (for 300B", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 313, 505, 326 ], "spans": [ { "bbox": [ 106, 313, 505, 326 ], "score": 1.0, "content": "tokens) and an updated version of WebMath (Polu & Sutskever, 2020) (for 72B tokens) whose mix is", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 325, 211, 336 ], "spans": [ { "bbox": [ 105, 325, 211, 336 ], "score": 1.0, "content": "presented in Appendix C.", "type": "text" } ], "index": 17 } ], "index": 16, "bbox_fs": [ 105, 302, 505, 336 ] }, { "type": "title", "bbox": [ 108, 350, 230, 361 ], "lines": [ { "bbox": [ 106, 349, 231, 362 ], "spans": [ { "bbox": [ 106, 349, 231, 362 ], "score": 1.0, "content": "4.2 TRAINING OBJECTIVES", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 370, 505, 414 ], "lines": [ { "bbox": [ 105, 370, 506, 383 ], "spans": [ { "bbox": [ 105, 370, 506, 383 ], "score": 1.0, "content": "Proofstep objective The proofstep objective, introduced in Polu & Sutskever (2020), consists in", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 382, 506, 394 ], "spans": [ { "bbox": [ 105, 382, 506, 394 ], "score": 1.0, "content": "generating a PROOFSTEP (a Lean tactic) given a GOAL (a Lean tactic state). We also condition this", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 393, 506, 406 ], "spans": [ { "bbox": [ 105, 393, 506, 406 ], "score": 1.0, "content": "objective on the current DECLARATION (a Lean theorem name), which remains the same throughout a", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 404, 424, 415 ], "spans": [ { "bbox": [ 105, 404, 424, 415 ], "score": 1.0, "content": "proof search: DECL GOAL PROOFSTEP .", "type": "text" } ], "index": 22 } ], "index": 20.5, "bbox_fs": [ 105, 370, 506, 415 ] }, { "type": "text", "bbox": [ 107, 420, 505, 520 ], "lines": [ { "bbox": [ 106, 420, 505, 432 ], "spans": [ { "bbox": [ 106, 420, 505, 432 ], "score": 1.0, "content": "The rationale for conditioning on the declaration name is to hint our models on the position of the", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 430, 505, 444 ], "spans": [ { "bbox": [ 105, 430, 505, 444 ], "score": 1.0, "content": "current declaration in the mathlib library. It can be considered as a weak proxy signal for the large", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 442, 505, 454 ], "spans": [ { "bbox": [ 105, 442, 505, 454 ], "score": 1.0, "content": "amount of information not shown to the model (the full environment consisting of the available", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 453, 505, 465 ], "spans": [ { "bbox": [ 106, 453, 505, 465 ], "score": 1.0, "content": "imports and currently open declarations such as module names, notations, declared instances, ...). The", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 463, 507, 477 ], "spans": [ { "bbox": [ 105, 463, 507, 477 ], "score": 1.0, "content": "declaration name lets models at least in principle memorize and then retrieve some of that information,", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 476, 505, 487 ], "spans": [ { "bbox": [ 106, 476, 505, 487 ], "score": 1.0, "content": "knowing that lean-gym errors if a theorem or definition that is not available in the environment", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 487, 505, 498 ], "spans": [ { "bbox": [ 106, 487, 505, 498 ], "score": 1.0, "content": "associated with the current declaration is used by tactics generated by our models. Also note that", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 497, 505, 510 ], "spans": [ { "bbox": [ 105, 497, 367, 510 ], "score": 1.0, "content": "conversely to Polu & Sutskever (2020) and like Han et al. (2022)", "type": "text" }, { "bbox": [ 367, 498, 399, 508 ], "score": 0.49, "content": "{ < } G O A L >", "type": "inline_equation" }, { "bbox": [ 399, 497, 505, 510 ], "score": 1.0, "content": "is not necessarily a single", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 508, 385, 521 ], "spans": [ { "bbox": [ 105, 508, 385, 521 ], "score": 1.0, "content": "goal but a Lean tactic state, which possibly comprises multiple goals.", "type": "text" } ], "index": 31 } ], "index": 27, "bbox_fs": [ 105, 420, 507, 521 ] }, { "type": "text", "bbox": [ 107, 525, 505, 568 ], "lines": [ { "bbox": [ 105, 524, 505, 538 ], "spans": [ { "bbox": [ 105, 524, 505, 538 ], "score": 1.0, "content": "Proofsize objective We depart from Polu & Sutskever (2020) and use a proofsize objective to", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 536, 505, 549 ], "spans": [ { "bbox": [ 105, 536, 505, 549 ], "score": 1.0, "content": "guide our proof searches, which consists in generating one token that represents a proof size", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 547, 505, 559 ], "spans": [ { "bbox": [ 105, 547, 473, 559 ], "score": 1.0, "content": "estimate bucket for the current goal (Lean tactic state): DECL GOAL", "type": "text" }, { "bbox": [ 473, 547, 505, 558 ], "score": 0.38, "content": "{ < } G O A L >", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 106, 559, 277, 569 ], "spans": [ { "bbox": [ 106, 559, 277, 569 ], "score": 1.0, "content": "PROOFSIZE ", "type": "text" } ], "index": 35 } ], "index": 33.5, "bbox_fs": [ 105, 524, 505, 569 ] }, { "type": "text", "bbox": [ 107, 574, 505, 619 ], "lines": [ { "bbox": [ 105, 574, 505, 587 ], "spans": [ { "bbox": [ 105, 574, 172, 587 ], "score": 1.0, "content": "For a given goal", "type": "text" }, { "bbox": [ 172, 577, 179, 586 ], "score": 0.71, "content": "g", "type": "inline_equation" }, { "bbox": [ 179, 574, 505, 587 ], "score": 1.0, "content": ", either the goal was proved as part of the proof search and we denote its proof size", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 586, 506, 599 ], "spans": [ { "bbox": [ 105, 586, 432, 599 ], "score": 1.0, "content": "(the number of tactic applications (compounded Lean tactics counting as one)) as", "type": "text" }, { "bbox": [ 433, 586, 456, 598 ], "score": 0.91, "content": "p s ( g )", "type": "inline_equation" }, { "bbox": [ 456, 586, 506, 599 ], "score": 1.0, "content": ", or the goal", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 597, 506, 609 ], "spans": [ { "bbox": [ 105, 597, 506, 609 ], "score": 1.0, "content": "was not proved in which case we assign the goal to a bucket that virtually represents \"infinite\" proof", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 608, 132, 620 ], "spans": [ { "bbox": [ 105, 608, 132, 620 ], "score": 1.0, "content": "sizes.", "type": "text" } ], "index": 39 } ], "index": 37.5, "bbox_fs": [ 105, 574, 506, 620 ] }, { "type": "text", "bbox": [ 107, 624, 505, 669 ], "lines": [ { "bbox": [ 105, 623, 505, 638 ], "spans": [ { "bbox": [ 105, 623, 186, 638 ], "score": 1.0, "content": "We use 11 buckets", "type": "text" }, { "bbox": [ 187, 625, 235, 635 ], "score": 0.88, "content": "B = 0 . . . 1 0", "type": "inline_equation" }, { "bbox": [ 236, 623, 379, 638 ], "score": 1.0, "content": "and compute the proofsize bucket", "type": "text" }, { "bbox": [ 379, 624, 397, 636 ], "score": 0.9, "content": "b ( g )", "type": "inline_equation" }, { "bbox": [ 397, 623, 442, 638 ], "score": 1.0, "content": "for a goal", "type": "text" }, { "bbox": [ 442, 627, 449, 636 ], "score": 0.76, "content": "g", "type": "inline_equation" }, { "bbox": [ 449, 623, 505, 638 ], "score": 1.0, "content": "by assigning", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 635, 505, 648 ], "spans": [ { "bbox": [ 105, 635, 505, 648 ], "score": 1.0, "content": "infinite proof sizes to bucket 0, all proof sizes over 20 to bucket 1 and linearly projecting proof sizes", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 646, 506, 659 ], "spans": [ { "bbox": [ 105, 646, 264, 659 ], "score": 1.0, "content": "lower than 20 on the remaining buckets", "type": "text" }, { "bbox": [ 264, 647, 298, 658 ], "score": 0.69, "content": "2 , . . . , 1 0", "type": "inline_equation" }, { "bbox": [ 298, 646, 506, 659 ], "score": 1.0, "content": "(10 being the bucket for the shortest proof sizes). In", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 658, 460, 669 ], "spans": [ { "bbox": [ 105, 658, 356, 669 ], "score": 1.0, "content": "practice, when training and sampling from the model, we map", "type": "text" }, { "bbox": [ 357, 658, 366, 667 ], "score": 0.8, "content": "B", "type": "inline_equation" }, { "bbox": [ 366, 658, 421, 669 ], "score": 1.0, "content": "to the tokens", "type": "text" }, { "bbox": [ 421, 658, 456, 668 ], "score": 0.72, "content": "\\therefore \\mathrm { A \\Omega } . . . \\mathrm { K \\Omega }", "type": "inline_equation" }, { "bbox": [ 456, 658, 460, 669 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 43 } ], "index": 41.5, "bbox_fs": [ 105, 623, 506, 669 ] }, { "type": "text", "bbox": [ 107, 674, 505, 732 ], "lines": [ { "bbox": [ 105, 674, 506, 687 ], "spans": [ { "bbox": [ 105, 674, 506, 687 ], "score": 1.0, "content": "To value goals as we run proof searches, we sample the proofsize bucket token and record the", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 684, 505, 699 ], "spans": [ { "bbox": [ 106, 684, 153, 699 ], "score": 1.0, "content": "probability", "type": "text" }, { "bbox": [ 153, 685, 176, 696 ], "score": 0.89, "content": "p _ { b } ( g )", "type": "inline_equation" }, { "bbox": [ 176, 684, 505, 699 ], "score": 1.0, "content": "for each viable bucket and use them to get a weighted average with the following", "type": "text" } ], "index": 45 }, { "bbox": [ 104, 694, 507, 713 ], "spans": [ { "bbox": [ 104, 694, 144, 713 ], "score": 1.0, "content": "formula:", "type": "text" }, { "bbox": [ 144, 696, 254, 711 ], "score": 0.92, "content": "\\begin{array} { r } { \\dot { v } ( \\dot { g } ) = \\frac { 1 } { \\# B } \\sum _ { b \\in B } p _ { b } ( g ) \\cdot b } \\end{array}", "type": "inline_equation" }, { "bbox": [ 255, 694, 403, 713 ], "score": 1.0, "content": ". As an example, if the model assigns", "type": "text" }, { "bbox": [ 403, 697, 433, 708 ], "score": 0.93, "content": "p _ { 0 } = 1", "type": "inline_equation" }, { "bbox": [ 433, 694, 462, 713 ], "score": 1.0, "content": "(hence", "type": "text" }, { "bbox": [ 462, 697, 501, 709 ], "score": 0.9, "content": "p _ { b \\neq 0 } = 0", "type": "inline_equation" }, { "bbox": [ 502, 694, 507, 713 ], "score": 1.0, "content": ")", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 708, 506, 723 ], "spans": [ { "bbox": [ 105, 708, 126, 723 ], "score": 1.0, "content": "then", "type": "text" }, { "bbox": [ 127, 709, 164, 721 ], "score": 0.91, "content": "v ( g ) = 0", "type": "inline_equation" }, { "bbox": [ 164, 708, 299, 723 ], "score": 1.0, "content": ". Conversely if the model assigns", "type": "text" }, { "bbox": [ 299, 710, 332, 721 ], "score": 0.9, "content": "p _ { 1 0 } = 1", "type": "inline_equation" }, { "bbox": [ 333, 708, 506, 723 ], "score": 1.0, "content": "(10 being the bucket for the shortest proof", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 719, 193, 733 ], "spans": [ { "bbox": [ 105, 719, 151, 733 ], "score": 1.0, "content": "sizes) then", "type": "text" }, { "bbox": [ 151, 721, 189, 732 ], "score": 0.92, "content": "v ( g ) = 1", "type": "inline_equation" }, { "bbox": [ 189, 719, 193, 733 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 48 } ], "index": 46, "bbox_fs": [ 104, 674, 507, 733 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 136, 177, 291, 255 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 79, 505, 168 ], "group_id": 0, "lines": [ { "bbox": [ 105, 80, 505, 92 ], "spans": [ { "bbox": [ 105, 80, 209, 92 ], "score": 1.0, "content": "Table 1: Performance of", "type": "text" }, { "bbox": [ 209, 81, 219, 91 ], "score": 0.89, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 220, 80, 238, 92 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 238, 81, 249, 91 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 249, 80, 362, 92 ], "score": 1.0, "content": "on mathlib-valid and miniF", "type": "text" }, { "bbox": [ 362, 81, 376, 91 ], "score": 0.29, "content": "_ { 2 F }", "type": "inline_equation" }, { "bbox": [ 376, 80, 505, 92 ], "score": 1.0, "content": "-valid compared to PACT Lean", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 91, 504, 104 ], "spans": [ { "bbox": [ 105, 91, 493, 104 ], "score": 1.0, "content": "GPT-f as reported in Han et al. (2022); Zheng et al. (2022). All models have the same architecture.", "type": "text" }, { "bbox": [ 493, 91, 504, 102 ], "score": 0.85, "content": "\\theta _ { 0 }", "type": "inline_equation" } ], "index": 1 }, { "bbox": [ 105, 102, 505, 115 ], "spans": [ { "bbox": [ 105, 102, 358, 115 ], "score": 1.0, "content": "is sampled using cumulative logprob priority best-first search.", "type": "text" }, { "bbox": [ 358, 102, 368, 113 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 369, 102, 505, 115 ], "score": 1.0, "content": "is sampled using best-first search", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 113, 505, 126 ], "spans": [ { "bbox": [ 105, 114, 322, 126 ], "score": 1.0, "content": "based on the proofsize objective. We report our setup", "type": "text" }, { "bbox": [ 322, 113, 357, 123 ], "score": 0.88, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 357, 114, 421, 126 ], "score": 1.0, "content": "expansions and", "type": "text" }, { "bbox": [ 421, 114, 446, 123 ], "score": 0.9, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 446, 114, 505, 126 ], "score": 1.0, "content": "tactic samples", "type": "text" } ], "index": 3 }, { "bbox": [ 104, 123, 506, 138 ], "spans": [ { "bbox": [ 104, 123, 490, 138 ], "score": 1.0, "content": "per expansions) as well as the setups used in Han et al. (2022); Zheng et al. (2022) (denoted as", "type": "text" }, { "bbox": [ 490, 124, 501, 136 ], "score": 0.84, "content": "\\theta _ { 0 } ^ { * }", "type": "inline_equation" }, { "bbox": [ 502, 123, 506, 138 ], "score": 1.0, "content": ")", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 134, 505, 149 ], "spans": [ { "bbox": [ 104, 134, 344, 149 ], "score": 1.0, "content": "to control for compute. We also report the performance of", "type": "text" }, { "bbox": [ 344, 136, 354, 146 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 354, 134, 505, 149 ], "score": 1.0, "content": "on mathlib-valid when trained using", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 146, 506, 158 ], "spans": [ { "bbox": [ 105, 146, 244, 158 ], "score": 1.0, "content": "the outcome objective (denoted as", "type": "text" }, { "bbox": [ 244, 146, 254, 158 ], "score": 0.85, "content": "\\theta _ { 1 } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 255, 146, 506, 158 ], "score": 1.0, "content": ") from Polu & Sutskever (2020) as an ablation of our proposed", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 158, 186, 169 ], "spans": [ { "bbox": [ 105, 158, 186, 169 ], "score": 1.0, "content": "proofsize objective.", "type": "text" } ], "index": 7 } ], "index": 3.5 }, { "type": "table_body", "bbox": [ 136, 177, 291, 255 ], "group_id": 0, "lines": [ { "bbox": [ 136, 177, 291, 255 ], "spans": [ { "bbox": [ 136, 177, 291, 255 ], "score": 0.754, "html": "
Modeldepass@1pass@8
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", "type": "table", "image_path": "c1e7d0150848dc75018e4de8236e20adca11b725bdc50a12624537f3abed71aa.jpg" } ] } ], "index": 13.0, "virtual_lines": [ { "bbox": [ 136, 177, 291, 190.0 ], "spans": [], "index": 8 }, { "bbox": [ 136, 190.0, 291, 203.0 ], "spans": [], "index": 10 }, { "bbox": [ 136, 203.0, 291, 216.0 ], "spans": [], "index": 12 }, { "bbox": [ 136, 216.0, 291, 229.0 ], "spans": [], "index": 14 }, { "bbox": [ 136, 229.0, 291, 242.0 ], "spans": [], "index": 16 }, { "bbox": [ 136, 242.0, 291, 255.0 ], "spans": [], "index": 18 } ] } ], "index": 8.25 }, { "type": "table", "bbox": [ 308, 177, 480, 255 ], "blocks": [ { "type": "table_body", "bbox": [ 308, 177, 480, 255 ], "group_id": 1, "lines": [ { "bbox": [ 308, 177, 480, 255 ], "spans": [ { "bbox": [ 308, 177, 480, 255 ], "score": 0.122, "html": "
Modeldepass@1pass@8
miniF2F-valid
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1281627.6%31.8%
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01512828.5%35.5%
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", "type": "table", "image_path": "67cd46cdf887810059756d0462d653e3112feb98e32f24bdc1dc32a94c207a1c.jpg" } ] } ], "index": 14.0, "virtual_lines": [ { "bbox": [ 308, 177, 480, 190.0 ], "spans": [], "index": 9 }, { "bbox": [ 308, 190.0, 480, 203.0 ], "spans": [], "index": 11 }, { "bbox": [ 308, 203.0, 480, 216.0 ], "spans": [], "index": 13 }, { "bbox": [ 308, 216.0, 480, 229.0 ], "spans": [], "index": 15 }, { "bbox": [ 308, 229.0, 480, 242.0 ], "spans": [], "index": 17 }, { "bbox": [ 308, 242.0, 480, 255.0 ], "spans": [], "index": 19 } ] } ], "index": 14.0 }, { "type": "text", "bbox": [ 106, 283, 505, 361 ], "lines": [ { "bbox": [ 105, 282, 505, 296 ], "spans": [ { "bbox": [ 105, 282, 505, 296 ], "score": 1.0, "content": "The rationale for using this proofsize objective instead of the outcome objective described in Polu", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 294, 505, 307 ], "spans": [ { "bbox": [ 105, 294, 505, 307 ], "score": 1.0, "content": "& Sutskever (2020) is that (i) it achieves better performance compared to the outcome objective", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 305, 506, 318 ], "spans": [ { "bbox": [ 105, 305, 506, 318 ], "score": 1.0, "content": "(see Table 1), and (ii) it prioritizes goals that potentially lead to shorter proofs during proof search,", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 316, 506, 329 ], "spans": [ { "bbox": [ 105, 316, 506, 329 ], "score": 1.0, "content": "creating an intrinsic incentive for the system to converge towards shorter proofs. Similarly to Polu &", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 326, 506, 340 ], "spans": [ { "bbox": [ 105, 326, 506, 340 ], "score": 1.0, "content": "Sutskever (2020) we favor this token-based approach to the introduction of a separate value head to", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 336, 505, 352 ], "spans": [ { "bbox": [ 105, 336, 505, 352 ], "score": 1.0, "content": "keep the overall architecture simple. This way the proofsize objective can be implemented by simply", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 348, 387, 363 ], "spans": [ { "bbox": [ 105, 348, 387, 363 ], "score": 1.0, "content": "augmenting the training dataset and without any architectural change.", "type": "text" } ], "index": 26 } ], "index": 23 }, { "type": "title", "bbox": [ 108, 378, 204, 389 ], "lines": [ { "bbox": [ 106, 377, 205, 390 ], "spans": [ { "bbox": [ 106, 377, 205, 390 ], "score": 1.0, "content": "4.3 BOOTSTRAPPING", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 108, 399, 504, 422 ], "lines": [ { "bbox": [ 105, 398, 505, 413 ], "spans": [ { "bbox": [ 105, 398, 505, 413 ], "score": 1.0, "content": "Bootstrapping consists in the steps required to train an initial model on both the proofstep objective", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 411, 217, 423 ], "spans": [ { "bbox": [ 106, 411, 217, 423 ], "score": 1.0, "content": "and the proofsize objective.", "type": "text" } ], "index": 29 } ], "index": 28.5 }, { "type": "text", "bbox": [ 107, 428, 505, 483 ], "lines": [ { "bbox": [ 106, 428, 505, 439 ], "spans": [ { "bbox": [ 106, 428, 505, 439 ], "score": 1.0, "content": "Given a pre-trained model on WebMath, we fine-tune it on the tactic dataset extracted from mathlib", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 438, 505, 451 ], "spans": [ { "bbox": [ 105, 438, 505, 451 ], "score": 1.0, "content": "as well as the proof artifacts dataset mix1 as described in Han et al. (2022). This initial model, which", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 449, 505, 462 ], "spans": [ { "bbox": [ 105, 449, 149, 462 ], "score": 1.0, "content": "we denote", "type": "text" }, { "bbox": [ 149, 450, 160, 461 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 160, 449, 505, 462 ], "score": 1.0, "content": "is solely trained on the proofstep objective. We use the validation splits of the tactic", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 460, 505, 473 ], "spans": [ { "bbox": [ 105, 460, 505, 473 ], "score": 1.0, "content": "and m1 datasets to early-stop training. Note that this is our only use of mathlib-valid to influence the", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 472, 262, 485 ], "spans": [ { "bbox": [ 106, 472, 262, 485 ], "score": 1.0, "content": "training process throughout this paper.", "type": "text" } ], "index": 34 } ], "index": 32 }, { "type": "text", "bbox": [ 107, 488, 505, 543 ], "lines": [ { "bbox": [ 105, 488, 506, 501 ], "spans": [ { "bbox": [ 105, 488, 311, 501 ], "score": 1.0, "content": "To generate data for the proofsize objective, we use", "type": "text" }, { "bbox": [ 311, 489, 321, 500 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 321, 488, 506, 501 ], "score": 1.0, "content": "to sample proofs for statements from mathlib-", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 499, 505, 512 ], "spans": [ { "bbox": [ 105, 499, 371, 512 ], "score": 1.0, "content": "train. For each statement from mathlib-train (25k) we attempt", "type": "text" }, { "bbox": [ 371, 500, 399, 510 ], "score": 0.89, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 400, 499, 505, 512 ], "score": 1.0, "content": "proof searches using the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 510, 506, 523 ], "spans": [ { "bbox": [ 105, 510, 506, 523 ], "score": 1.0, "content": "cumulative logprob priority search described in Polu & Sutskever (2020) (which does not require a", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 520, 505, 534 ], "spans": [ { "bbox": [ 105, 520, 224, 534 ], "score": 1.0, "content": "trained value function) using", "type": "text" }, { "bbox": [ 225, 522, 259, 532 ], "score": 0.89, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 259, 520, 324, 534 ], "score": 1.0, "content": "expansions and", "type": "text" }, { "bbox": [ 324, 522, 348, 532 ], "score": 0.89, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 349, 520, 505, 534 ], "score": 1.0, "content": "samples per expansion. We denote the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 531, 352, 546 ], "spans": [ { "bbox": [ 105, 531, 336, 546 ], "score": 1.0, "content": "set of successful proof searches created in this process as", "type": "text" }, { "bbox": [ 336, 533, 348, 543 ], "score": 0.88, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 348, 531, 352, 546 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 39 } ], "index": 37 }, { "type": "text", "bbox": [ 107, 549, 505, 594 ], "lines": [ { "bbox": [ 105, 548, 507, 563 ], "spans": [ { "bbox": [ 105, 548, 132, 563 ], "score": 1.0, "content": "Using", "type": "text" }, { "bbox": [ 132, 549, 144, 560 ], "score": 0.87, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 144, 548, 222, 563 ], "score": 1.0, "content": "we generate dataset", "type": "text" }, { "bbox": [ 223, 550, 236, 560 ], "score": 0.9, "content": "D _ { 0 }", "type": "inline_equation" }, { "bbox": [ 237, 548, 507, 563 ], "score": 1.0, "content": "by concatenating: (i) the initial tactic dataset (proofstep objective),", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 560, 506, 573 ], "spans": [ { "bbox": [ 105, 560, 365, 573 ], "score": 1.0, "content": "(ii) a deduplicated set of proofsteps extracted from the proofs in", "type": "text" }, { "bbox": [ 365, 560, 377, 571 ], "score": 0.88, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 377, 560, 506, 573 ], "score": 1.0, "content": "(proofstep objective) and (iii) a", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 570, 504, 584 ], "spans": [ { "bbox": [ 105, 570, 492, 584 ], "score": 1.0, "content": "deduplicated set of proofsize tuples (goals and proofsize) extracted from the full proof searches in", "type": "text" }, { "bbox": [ 492, 572, 504, 582 ], "score": 0.87, "content": "S _ { 0 }", "type": "inline_equation" } ], "index": 42 }, { "bbox": [ 105, 583, 192, 595 ], "spans": [ { "bbox": [ 105, 583, 192, 595 ], "score": 1.0, "content": "(proofsize objective).", "type": "text" } ], "index": 43 } ], "index": 41.5 }, { "type": "text", "bbox": [ 107, 599, 505, 643 ], "lines": [ { "bbox": [ 105, 599, 506, 612 ], "spans": [ { "bbox": [ 105, 599, 246, 612 ], "score": 1.0, "content": "Note that the full proof searches in", "type": "text" }, { "bbox": [ 247, 600, 258, 610 ], "score": 0.88, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 259, 599, 506, 612 ], "score": 1.0, "content": "include goals that are visited but eventually remain unproved,", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 609, 506, 623 ], "spans": [ { "bbox": [ 105, 609, 506, 623 ], "score": 1.0, "content": "which provides useful negative examples for the trained value function (even if these negatives may", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 621, 506, 633 ], "spans": [ { "bbox": [ 106, 622, 459, 633 ], "score": 1.0, "content": "include provable goals that simply were not prioritized by the search). Also note that", "type": "text" }, { "bbox": [ 459, 621, 471, 632 ], "score": 0.89, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 471, 622, 506, 633 ], "score": 1.0, "content": "doesn’t", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 631, 226, 644 ], "spans": [ { "bbox": [ 105, 631, 226, 644 ], "score": 1.0, "content": "include failed proof searches.", "type": "text" } ], "index": 47 } ], "index": 45.5 }, { "type": "text", "bbox": [ 107, 649, 505, 693 ], "lines": [ { "bbox": [ 105, 648, 505, 661 ], "spans": [ { "bbox": [ 105, 648, 159, 661 ], "score": 1.0, "content": "We fine-tune", "type": "text" }, { "bbox": [ 159, 649, 169, 660 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 170, 648, 183, 661 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 183, 649, 197, 660 ], "score": 0.89, "content": "D _ { 0 }", "type": "inline_equation" }, { "bbox": [ 198, 648, 505, 661 ], "score": 1.0, "content": "for exactly one epoch (no use of validation data for early-stopping) to obtain", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 660, 505, 672 ], "spans": [ { "bbox": [ 106, 660, 175, 672 ], "score": 1.0, "content": "our initial model", "type": "text" }, { "bbox": [ 175, 660, 185, 671 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 186, 660, 453, 672 ], "score": 1.0, "content": "trained on both the proofstep objective and the proofsize objective.", "type": "text" }, { "bbox": [ 453, 660, 464, 671 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 464, 660, 505, 672 ], "score": 1.0, "content": "is used in", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 671, 505, 682 ], "spans": [ { "bbox": [ 105, 671, 419, 682 ], "score": 1.0, "content": "our expert iteration setup as base model to fine-tune from at each iteration, and", "type": "text" }, { "bbox": [ 419, 671, 429, 682 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 430, 671, 505, 682 ], "score": 1.0, "content": "is our first iterated", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 682, 366, 694 ], "spans": [ { "bbox": [ 106, 682, 366, 694 ], "score": 1.0, "content": "model or mathlib bootstrapped model trained on both objectives.", "type": "text" } ], "index": 51 } ], "index": 49.5 }, { "type": "text", "bbox": [ 108, 699, 505, 732 ], "lines": [ { "bbox": [ 106, 698, 505, 711 ], "spans": [ { "bbox": [ 106, 698, 251, 711 ], "score": 1.0, "content": "We report in Table 1 the pass rates of", "type": "text" }, { "bbox": [ 252, 700, 261, 710 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 262, 698, 279, 711 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 279, 700, 289, 710 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 289, 698, 505, 711 ], "score": 1.0, "content": "on mathlib-valid and miniF2F-valid and compare with", "type": "text" } ], "index": 52 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "previously reported pass rates for equivalent amounts of compute. As reported in Polu & Sutskever", "type": "text" } ], "index": 53 }, { "bbox": [ 106, 720, 506, 733 ], "spans": [ { "bbox": [ 106, 720, 426, 733 ], "score": 1.0, "content": "(2020), training a value function to guide search greatly improves the pass rates of", "type": "text" }, { "bbox": [ 426, 721, 436, 732 ], "score": 0.89, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 437, 720, 506, 733 ], "score": 1.0, "content": "on mathlib-valid.", "type": "text" } ], "index": 54 } ], "index": 53 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 752, 308, 759 ], "lines": [] } ], "para_blocks": [ { "type": "table", "bbox": [ 136, 177, 291, 255 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 79, 505, 168 ], "group_id": 0, "lines": [ { "bbox": [ 105, 80, 505, 92 ], "spans": [ { "bbox": [ 105, 80, 209, 92 ], "score": 1.0, "content": "Table 1: Performance of", "type": "text" }, { "bbox": [ 209, 81, 219, 91 ], "score": 0.89, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 220, 80, 238, 92 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 238, 81, 249, 91 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 249, 80, 362, 92 ], "score": 1.0, "content": "on mathlib-valid and miniF", "type": "text" }, { "bbox": [ 362, 81, 376, 91 ], "score": 0.29, "content": "_ { 2 F }", "type": "inline_equation" }, { "bbox": [ 376, 80, 505, 92 ], "score": 1.0, "content": "-valid compared to PACT Lean", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 91, 504, 104 ], "spans": [ { "bbox": [ 105, 91, 493, 104 ], "score": 1.0, "content": "GPT-f as reported in Han et al. (2022); Zheng et al. (2022). All models have the same architecture.", "type": "text" }, { "bbox": [ 493, 91, 504, 102 ], "score": 0.85, "content": "\\theta _ { 0 }", "type": "inline_equation" } ], "index": 1 }, { "bbox": [ 105, 102, 505, 115 ], "spans": [ { "bbox": [ 105, 102, 358, 115 ], "score": 1.0, "content": "is sampled using cumulative logprob priority best-first search.", "type": "text" }, { "bbox": [ 358, 102, 368, 113 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 369, 102, 505, 115 ], "score": 1.0, "content": "is sampled using best-first search", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 113, 505, 126 ], "spans": [ { "bbox": [ 105, 114, 322, 126 ], "score": 1.0, "content": "based on the proofsize objective. We report our setup", "type": "text" }, { "bbox": [ 322, 113, 357, 123 ], "score": 0.88, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 357, 114, 421, 126 ], "score": 1.0, "content": "expansions and", "type": "text" }, { "bbox": [ 421, 114, 446, 123 ], "score": 0.9, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 446, 114, 505, 126 ], "score": 1.0, "content": "tactic samples", "type": "text" } ], "index": 3 }, { "bbox": [ 104, 123, 506, 138 ], "spans": [ { "bbox": [ 104, 123, 490, 138 ], "score": 1.0, "content": "per expansions) as well as the setups used in Han et al. (2022); Zheng et al. 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Similarly to Polu &", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 326, 506, 340 ], "spans": [ { "bbox": [ 105, 326, 506, 340 ], "score": 1.0, "content": "Sutskever (2020) we favor this token-based approach to the introduction of a separate value head to", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 336, 505, 352 ], "spans": [ { "bbox": [ 105, 336, 505, 352 ], "score": 1.0, "content": "keep the overall architecture simple. This way the proofsize objective can be implemented by simply", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 348, 387, 363 ], "spans": [ { "bbox": [ 105, 348, 387, 363 ], "score": 1.0, "content": "augmenting the training dataset and without any architectural change.", "type": "text" } ], "index": 26 } ], "index": 23, "bbox_fs": [ 105, 282, 506, 363 ] }, { "type": "title", "bbox": [ 108, 378, 204, 389 ], "lines": [ { "bbox": [ 106, 377, 205, 390 ], "spans": [ { "bbox": [ 106, 377, 205, 390 ], "score": 1.0, "content": "4.3 BOOTSTRAPPING", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 108, 399, 504, 422 ], "lines": [ { "bbox": [ 105, 398, 505, 413 ], "spans": [ { "bbox": [ 105, 398, 505, 413 ], "score": 1.0, "content": "Bootstrapping consists in the steps required to train an initial model on both the proofstep objective", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 411, 217, 423 ], "spans": [ { "bbox": [ 106, 411, 217, 423 ], "score": 1.0, "content": "and the proofsize objective.", "type": "text" } ], "index": 29 } ], "index": 28.5, "bbox_fs": [ 105, 398, 505, 423 ] }, { "type": "text", "bbox": [ 107, 428, 505, 483 ], "lines": [ { "bbox": [ 106, 428, 505, 439 ], "spans": [ { "bbox": [ 106, 428, 505, 439 ], "score": 1.0, "content": "Given a pre-trained model on WebMath, we fine-tune it on the tactic dataset extracted from mathlib", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 438, 505, 451 ], "spans": [ { "bbox": [ 105, 438, 505, 451 ], "score": 1.0, "content": "as well as the proof artifacts dataset mix1 as described in Han et al. (2022). This initial model, which", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 449, 505, 462 ], "spans": [ { "bbox": [ 105, 449, 149, 462 ], "score": 1.0, "content": "we denote", "type": "text" }, { "bbox": [ 149, 450, 160, 461 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 160, 449, 505, 462 ], "score": 1.0, "content": "is solely trained on the proofstep objective. We use the validation splits of the tactic", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 460, 505, 473 ], "spans": [ { "bbox": [ 105, 460, 505, 473 ], "score": 1.0, "content": "and m1 datasets to early-stop training. Note that this is our only use of mathlib-valid to influence the", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 472, 262, 485 ], "spans": [ { "bbox": [ 106, 472, 262, 485 ], "score": 1.0, "content": "training process throughout this paper.", "type": "text" } ], "index": 34 } ], "index": 32, "bbox_fs": [ 105, 428, 505, 485 ] }, { "type": "text", "bbox": [ 107, 488, 505, 543 ], "lines": [ { "bbox": [ 105, 488, 506, 501 ], "spans": [ { "bbox": [ 105, 488, 311, 501 ], "score": 1.0, "content": "To generate data for the proofsize objective, we use", "type": "text" }, { "bbox": [ 311, 489, 321, 500 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 321, 488, 506, 501 ], "score": 1.0, "content": "to sample proofs for statements from mathlib-", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 499, 505, 512 ], "spans": [ { "bbox": [ 105, 499, 371, 512 ], "score": 1.0, "content": "train. For each statement from mathlib-train (25k) we attempt", "type": "text" }, { "bbox": [ 371, 500, 399, 510 ], "score": 0.89, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 400, 499, 505, 512 ], "score": 1.0, "content": "proof searches using the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 510, 506, 523 ], "spans": [ { "bbox": [ 105, 510, 506, 523 ], "score": 1.0, "content": "cumulative logprob priority search described in Polu & Sutskever (2020) (which does not require a", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 520, 505, 534 ], "spans": [ { "bbox": [ 105, 520, 224, 534 ], "score": 1.0, "content": "trained value function) using", "type": "text" }, { "bbox": [ 225, 522, 259, 532 ], "score": 0.89, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 259, 520, 324, 534 ], "score": 1.0, "content": "expansions and", "type": "text" }, { "bbox": [ 324, 522, 348, 532 ], "score": 0.89, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 349, 520, 505, 534 ], "score": 1.0, "content": "samples per expansion. We denote the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 531, 352, 546 ], "spans": [ { "bbox": [ 105, 531, 336, 546 ], "score": 1.0, "content": "set of successful proof searches created in this process as", "type": "text" }, { "bbox": [ 336, 533, 348, 543 ], "score": 0.88, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 348, 531, 352, 546 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 39 } ], "index": 37, "bbox_fs": [ 105, 488, 506, 546 ] }, { "type": "text", "bbox": [ 107, 549, 505, 594 ], "lines": [ { "bbox": [ 105, 548, 507, 563 ], "spans": [ { "bbox": [ 105, 548, 132, 563 ], "score": 1.0, "content": "Using", "type": "text" }, { "bbox": [ 132, 549, 144, 560 ], "score": 0.87, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 144, 548, 222, 563 ], "score": 1.0, "content": "we generate dataset", "type": "text" }, { "bbox": [ 223, 550, 236, 560 ], "score": 0.9, "content": "D _ { 0 }", "type": "inline_equation" }, { "bbox": [ 237, 548, 507, 563 ], "score": 1.0, "content": "by concatenating: (i) the initial tactic dataset (proofstep objective),", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 560, 506, 573 ], "spans": [ { "bbox": [ 105, 560, 365, 573 ], "score": 1.0, "content": "(ii) a deduplicated set of proofsteps extracted from the proofs in", "type": "text" }, { "bbox": [ 365, 560, 377, 571 ], "score": 0.88, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 377, 560, 506, 573 ], "score": 1.0, "content": "(proofstep objective) and (iii) a", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 570, 504, 584 ], "spans": [ { "bbox": [ 105, 570, 492, 584 ], "score": 1.0, "content": "deduplicated set of proofsize tuples (goals and proofsize) extracted from the full proof searches in", "type": "text" }, { "bbox": [ 492, 572, 504, 582 ], "score": 0.87, "content": "S _ { 0 }", "type": "inline_equation" } ], "index": 42 }, { "bbox": [ 105, 583, 192, 595 ], "spans": [ { "bbox": [ 105, 583, 192, 595 ], "score": 1.0, "content": "(proofsize objective).", "type": "text" } ], "index": 43 } ], "index": 41.5, "bbox_fs": [ 105, 548, 507, 595 ] }, { "type": "text", "bbox": [ 107, 599, 505, 643 ], "lines": [ { "bbox": [ 105, 599, 506, 612 ], "spans": [ { "bbox": [ 105, 599, 246, 612 ], "score": 1.0, "content": "Note that the full proof searches in", "type": "text" }, { "bbox": [ 247, 600, 258, 610 ], "score": 0.88, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 259, 599, 506, 612 ], "score": 1.0, "content": "include goals that are visited but eventually remain unproved,", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 609, 506, 623 ], "spans": [ { "bbox": [ 105, 609, 506, 623 ], "score": 1.0, "content": "which provides useful negative examples for the trained value function (even if these negatives may", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 621, 506, 633 ], "spans": [ { "bbox": [ 106, 622, 459, 633 ], "score": 1.0, "content": "include provable goals that simply were not prioritized by the search). Also note that", "type": "text" }, { "bbox": [ 459, 621, 471, 632 ], "score": 0.89, "content": "S _ { 0 }", "type": "inline_equation" }, { "bbox": [ 471, 622, 506, 633 ], "score": 1.0, "content": "doesn’t", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 631, 226, 644 ], "spans": [ { "bbox": [ 105, 631, 226, 644 ], "score": 1.0, "content": "include failed proof searches.", "type": "text" } ], "index": 47 } ], "index": 45.5, "bbox_fs": [ 105, 599, 506, 644 ] }, { "type": "text", "bbox": [ 107, 649, 505, 693 ], "lines": [ { "bbox": [ 105, 648, 505, 661 ], "spans": [ { "bbox": [ 105, 648, 159, 661 ], "score": 1.0, "content": "We fine-tune", "type": "text" }, { "bbox": [ 159, 649, 169, 660 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 170, 648, 183, 661 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 183, 649, 197, 660 ], "score": 0.89, "content": "D _ { 0 }", "type": "inline_equation" }, { "bbox": [ 198, 648, 505, 661 ], "score": 1.0, "content": "for exactly one epoch (no use of validation data for early-stopping) to obtain", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 660, 505, 672 ], "spans": [ { "bbox": [ 106, 660, 175, 672 ], "score": 1.0, "content": "our initial model", "type": "text" }, { "bbox": [ 175, 660, 185, 671 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 186, 660, 453, 672 ], "score": 1.0, "content": "trained on both the proofstep objective and the proofsize objective.", "type": "text" }, { "bbox": [ 453, 660, 464, 671 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 464, 660, 505, 672 ], "score": 1.0, "content": "is used in", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 671, 505, 682 ], "spans": [ { "bbox": [ 105, 671, 419, 682 ], "score": 1.0, "content": "our expert iteration setup as base model to fine-tune from at each iteration, and", "type": "text" }, { "bbox": [ 419, 671, 429, 682 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 430, 671, 505, 682 ], "score": 1.0, "content": "is our first iterated", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 682, 366, 694 ], "spans": [ { "bbox": [ 106, 682, 366, 694 ], "score": 1.0, "content": "model or mathlib bootstrapped model trained on both objectives.", "type": "text" } ], "index": 51 } ], "index": 49.5, "bbox_fs": [ 105, 648, 505, 694 ] }, { "type": "text", "bbox": [ 108, 699, 505, 732 ], "lines": [ { "bbox": [ 106, 698, 505, 711 ], "spans": [ { "bbox": [ 106, 698, 251, 711 ], "score": 1.0, "content": "We report in Table 1 the pass rates of", "type": "text" }, { "bbox": [ 252, 700, 261, 710 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 262, 698, 279, 711 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 279, 700, 289, 710 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 289, 698, 505, 711 ], "score": 1.0, "content": "on mathlib-valid and miniF2F-valid and compare with", "type": "text" } ], "index": 52 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "previously reported pass rates for equivalent amounts of compute. As reported in Polu & Sutskever", "type": "text" } ], "index": 53 }, { "bbox": [ 106, 720, 506, 733 ], "spans": [ { "bbox": [ 106, 720, 426, 733 ], "score": 1.0, "content": "(2020), training a value function to guide search greatly improves the pass rates of", "type": "text" }, { "bbox": [ 426, 721, 436, 732 ], "score": 0.89, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 437, 720, 506, 733 ], "score": 1.0, "content": "on mathlib-valid.", "type": "text" } ], "index": 54 } ], "index": 53, "bbox_fs": [ 105, 698, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 137 ], "lines": [ { "bbox": [ 106, 83, 505, 95 ], "spans": [ { "bbox": [ 106, 83, 229, 95 ], "score": 1.0, "content": "Interestingly, the gap between", "type": "text" }, { "bbox": [ 230, 83, 240, 93 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 241, 83, 258, 95 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 258, 83, 269, 93 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 269, 83, 306, 95 ], "score": 1.0, "content": "on miniF", "type": "text" }, { "bbox": [ 306, 83, 320, 93 ], "score": 0.28, "content": "{ } ^ { 7 2 F }", "type": "inline_equation" }, { "bbox": [ 320, 83, 505, 95 ], "score": 1.0, "content": "-valid is not as significant, demonstrating that", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 106 ], "spans": [ { "bbox": [ 105, 93, 506, 106 ], "score": 1.0, "content": "training a value function on proofs sampled from mathlib-train has limited transfer to miniF2F-valid.", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 505, 117 ], "score": 1.0, "content": "The main differences with Zheng et al. (2022), potentially explaining the gap on miniF2F-valid", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 114, 506, 128 ], "spans": [ { "bbox": [ 107, 115, 137, 127 ], "score": 0.84, "content": "( 2 7 . 6 \\%", "type": "inline_equation" }, { "bbox": [ 137, 114, 149, 128 ], "score": 1.0, "content": "vs", "type": "text" }, { "bbox": [ 150, 115, 178, 126 ], "score": 0.86, "content": "2 3 . 9 \\%", "type": "inline_equation" }, { "bbox": [ 178, 114, 506, 128 ], "score": 1.0, "content": "), consists in the new pre-training described in Section 4.1 as well as the use of a", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 127, 397, 139 ], "spans": [ { "bbox": [ 106, 127, 397, 139 ], "score": 1.0, "content": "more recent mathlib checkpoint for the mix1, mix2 and tactic datasets.", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "title", "bbox": [ 108, 153, 287, 164 ], "lines": [ { "bbox": [ 105, 153, 289, 165 ], "spans": [ { "bbox": [ 105, 153, 289, 165 ], "score": 1.0, "content": "4.4 ITERATED SAMPLING AND TRAINING", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 107, 174, 505, 218 ], "lines": [ { "bbox": [ 105, 174, 505, 187 ], "spans": [ { "bbox": [ 105, 174, 387, 187 ], "score": 1.0, "content": "Our expert iteration process takes as input: (i) a set of formal statements", "type": "text" }, { "bbox": [ 387, 175, 398, 185 ], "score": 0.82, "content": "S t", "type": "inline_equation" }, { "bbox": [ 398, 174, 456, 187 ], "score": 1.0, "content": ", (ii) a function", "type": "text" }, { "bbox": [ 456, 174, 505, 185 ], "score": 0.9, "content": "a : S t \\mathbb { N }", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 105, 185, 505, 198 ], "spans": [ { "bbox": [ 105, 185, 505, 198 ], "score": 1.0, "content": "indicating the number of proof search attempts to run per statement at each iteration, (iii) a base", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 196, 505, 208 ], "spans": [ { "bbox": [ 106, 197, 133, 208 ], "score": 1.0, "content": "model", "type": "text" }, { "bbox": [ 133, 197, 144, 207 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 144, 197, 432, 208 ], "score": 1.0, "content": "to fine-tune from at each iteration, and (iv) a mathlib bootstrapped model", "type": "text" }, { "bbox": [ 432, 196, 442, 207 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 442, 197, 505, 208 ], "score": 1.0, "content": "trained on both", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 208, 506, 219 ], "spans": [ { "bbox": [ 106, 208, 506, 219 ], "score": 1.0, "content": "objectives. A high-level illustration of the iterated sampling and training is available in Appendix E.", "type": "text" } ], "index": 9 } ], "index": 7.5 }, { "type": "text", "bbox": [ 107, 223, 505, 290 ], "lines": [ { "bbox": [ 106, 224, 505, 236 ], "spans": [ { "bbox": [ 106, 224, 163, 236 ], "score": 1.0, "content": "Each iteration", "type": "text" }, { "bbox": [ 163, 225, 170, 234 ], "score": 0.8, "content": "k", "type": "inline_equation" }, { "bbox": [ 171, 224, 379, 236 ], "score": 1.0, "content": "consists in sampling proof searches for statements in", "type": "text" }, { "bbox": [ 379, 225, 390, 234 ], "score": 0.8, "content": "S t", "type": "inline_equation" }, { "bbox": [ 390, 224, 414, 236 ], "score": 1.0, "content": "using", "type": "text" }, { "bbox": [ 415, 224, 425, 235 ], "score": 0.88, "content": "\\theta _ { k }", "type": "inline_equation" }, { "bbox": [ 425, 224, 505, 236 ], "score": 1.0, "content": ", filtering successful", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 235, 505, 247 ], "spans": [ { "bbox": [ 105, 235, 168, 247 ], "score": 1.0, "content": "proof searches", "type": "text" }, { "bbox": [ 169, 235, 181, 246 ], "score": 0.88, "content": "S _ { k }", "type": "inline_equation" }, { "bbox": [ 181, 235, 282, 247 ], "score": 1.0, "content": "to extract a new dataset", "type": "text" }, { "bbox": [ 282, 235, 297, 246 ], "score": 0.89, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 297, 235, 366, 247 ], "score": 1.0, "content": ", and fine-tuning", "type": "text" }, { "bbox": [ 366, 235, 376, 246 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 377, 235, 439, 247 ], "score": 1.0, "content": "on it to obtain", "type": "text" }, { "bbox": [ 439, 235, 460, 247 ], "score": 0.91, "content": "\\theta _ { k + 1 }", "type": "inline_equation" }, { "bbox": [ 460, 235, 505, 247 ], "score": 1.0, "content": ", on which", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 246, 505, 258 ], "spans": [ { "bbox": [ 105, 246, 297, 258 ], "score": 1.0, "content": "we can iterate. To sample proof searches from", "type": "text" }, { "bbox": [ 297, 246, 309, 256 ], "score": 0.81, "content": "S t", "type": "inline_equation" }, { "bbox": [ 309, 246, 495, 258 ], "score": 1.0, "content": "we use the best-first search described in Polu", "type": "text" }, { "bbox": [ 496, 246, 505, 256 ], "score": 0.27, "content": "\\&", "type": "inline_equation" } ], "index": 12 }, { "bbox": [ 105, 257, 505, 269 ], "spans": [ { "bbox": [ 105, 257, 422, 269 ], "score": 1.0, "content": "Sutskever (2020) with the value function described in Section 4.2. We attempt", "type": "text" }, { "bbox": [ 423, 259, 429, 267 ], "score": 0.66, "content": "a", "type": "inline_equation" }, { "bbox": [ 429, 257, 505, 269 ], "score": 1.0, "content": "proof searches for", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 268, 505, 281 ], "spans": [ { "bbox": [ 105, 268, 167, 281 ], "score": 1.0, "content": "each statement", "type": "text" }, { "bbox": [ 167, 268, 207, 280 ], "score": 0.92, "content": "s ( s \\in S t )", "type": "inline_equation" }, { "bbox": [ 208, 268, 228, 281 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 229, 268, 264, 278 ], "score": 0.9, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 264, 268, 327, 281 ], "score": 1.0, "content": "expansions and", "type": "text" }, { "bbox": [ 327, 268, 352, 278 ], "score": 0.9, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 352, 268, 505, 281 ], "score": 1.0, "content": "samples per expansion. We denote the", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 279, 319, 291 ], "spans": [ { "bbox": [ 105, 279, 284, 291 ], "score": 1.0, "content": "set of successful proof searches for iteration", "type": "text" }, { "bbox": [ 284, 280, 291, 289 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 292, 279, 303, 291 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 303, 279, 315, 290 ], "score": 0.89, "content": "S _ { k }", "type": "inline_equation" }, { "bbox": [ 315, 279, 319, 291 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 12.5 }, { "type": "text", "bbox": [ 106, 295, 505, 343 ], "lines": [ { "bbox": [ 106, 295, 506, 309 ], "spans": [ { "bbox": [ 106, 295, 133, 309 ], "score": 1.0, "content": "Using", "type": "text" }, { "bbox": [ 134, 296, 146, 307 ], "score": 0.87, "content": "S _ { k }", "type": "inline_equation" }, { "bbox": [ 146, 295, 235, 309 ], "score": 1.0, "content": "we generate datasets", "type": "text" }, { "bbox": [ 235, 297, 249, 307 ], "score": 0.92, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 250, 295, 506, 309 ], "score": 1.0, "content": "by concatenating: (i) the initial tactic dataset (proofstep ob-", "type": "text" } ], "index": 16 }, { "bbox": [ 104, 304, 506, 322 ], "spans": [ { "bbox": [ 104, 304, 412, 322 ], "score": 1.0, "content": "jective), (ii) a deduplicated set of proofsteps extracted from the proofs in", "type": "text" }, { "bbox": [ 412, 307, 459, 320 ], "score": 0.92, "content": "\\textstyle \\bigcup _ { 1 \\leq i \\leq k } { \\bar { S } } _ { k }", "type": "inline_equation" }, { "bbox": [ 460, 304, 506, 322 ], "score": 1.0, "content": "(proofstep", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 318, 505, 332 ], "spans": [ { "bbox": [ 105, 318, 505, 332 ], "score": 1.0, "content": "objective), and (iii) a deduplicated set of proofsize tuples (goals and proofsize) extracted from the", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 329, 328, 344 ], "spans": [ { "bbox": [ 104, 329, 193, 344 ], "score": 1.0, "content": "full proof searches in", "type": "text" }, { "bbox": [ 193, 330, 240, 344 ], "score": 0.93, "content": "\\textstyle \\bigcup _ { 1 \\leq i \\leq k } S _ { k }", "type": "inline_equation" }, { "bbox": [ 240, 329, 328, 344 ], "score": 1.0, "content": "(proofsize objective).", "type": "text" } ], "index": 19 } ], "index": 17.5 }, { "type": "text", "bbox": [ 107, 347, 505, 425 ], "lines": [ { "bbox": [ 105, 347, 505, 360 ], "spans": [ { "bbox": [ 105, 347, 505, 360 ], "score": 1.0, "content": "We use a global deduplication across iterations for both proofsteps and proofsize tuples which", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 358, 506, 371 ], "spans": [ { "bbox": [ 105, 358, 506, 371 ], "score": 1.0, "content": "we found to be important to maintain the stability of the expert iteration procedure. This global", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 369, 505, 383 ], "spans": [ { "bbox": [ 105, 369, 505, 383 ], "score": 1.0, "content": "deduplication is somewhat equivalent for each statement to growing a unique proof tree by aggregating", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 381, 505, 392 ], "spans": [ { "bbox": [ 106, 381, 505, 392 ], "score": 1.0, "content": "all the proof searches that have been run for it across iterations. This virtual proof tree accumulates a", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 392, 506, 405 ], "spans": [ { "bbox": [ 105, 392, 506, 405 ], "score": 1.0, "content": "growing number of positive proof paths and visited goals that remain unproven. We use these goals", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 403, 505, 415 ], "spans": [ { "bbox": [ 105, 403, 505, 415 ], "score": 1.0, "content": "as negative examples for the proofsize objective, labeling them with an infinite proofsize. Positive", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 414, 424, 426 ], "spans": [ { "bbox": [ 106, 414, 424, 426 ], "score": 1.0, "content": "goals are deduplicated keeping the minimum proof sizes across proof searches.", "type": "text" } ], "index": 26 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 430, 505, 485 ], "lines": [ { "bbox": [ 105, 429, 506, 443 ], "spans": [ { "bbox": [ 105, 429, 138, 443 ], "score": 1.0, "content": "Finally", "type": "text" }, { "bbox": [ 138, 431, 149, 442 ], "score": 0.88, "content": "\\theta _ { k }", "type": "inline_equation" }, { "bbox": [ 149, 429, 257, 443 ], "score": 1.0, "content": "is obtained by fine-tuning", "type": "text" }, { "bbox": [ 257, 431, 267, 442 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 268, 429, 372, 443 ], "score": 1.0, "content": "for exactly one epoch on", "type": "text" }, { "bbox": [ 372, 431, 386, 442 ], "score": 0.89, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 386, 429, 506, 443 ], "score": 1.0, "content": ". Note that the initial tactic", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 440, 505, 454 ], "spans": [ { "bbox": [ 105, 440, 215, 454 ], "score": 1.0, "content": "dataset is included in each", "type": "text" }, { "bbox": [ 215, 442, 229, 452 ], "score": 0.88, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 230, 440, 264, 454 ], "score": 1.0, "content": ", despite", "type": "text" }, { "bbox": [ 265, 442, 275, 453 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 275, 440, 505, 454 ], "score": 1.0, "content": "being already trained on it (along with mix1). We found", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 453, 505, 464 ], "spans": [ { "bbox": [ 106, 453, 505, 464 ], "score": 1.0, "content": "this repetition to be beneficial overall (as it adds the mathlib extracted proofsteps to our deduplicated", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 463, 506, 476 ], "spans": [ { "bbox": [ 105, 463, 506, 476 ], "score": 1.0, "content": "per statements virtual proof trees) despite it leading to a slight overfit on the tactic dataset in terms", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 473, 180, 486 ], "spans": [ { "bbox": [ 105, 473, 180, 486 ], "score": 1.0, "content": "of validation loss.", "type": "text" } ], "index": 31 } ], "index": 29 }, { "type": "title", "bbox": [ 109, 501, 284, 512 ], "lines": [ { "bbox": [ 106, 500, 286, 514 ], "spans": [ { "bbox": [ 106, 500, 286, 514 ], "score": 1.0, "content": "4.5 EXPERT ITERATION ON mathlib-train", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 107, 522, 505, 588 ], "lines": [ { "bbox": [ 105, 522, 505, 535 ], "spans": [ { "bbox": [ 105, 522, 234, 535 ], "score": 1.0, "content": "In this section we propose to set", "type": "text" }, { "bbox": [ 234, 523, 245, 532 ], "score": 0.83, "content": "S t", "type": "inline_equation" }, { "bbox": [ 245, 522, 505, 535 ], "score": 1.0, "content": "to the statements in mathlib-train, run our expert iteration process", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 533, 505, 545 ], "spans": [ { "bbox": [ 106, 533, 505, 545 ], "score": 1.0, "content": "with it and report performance on both mathlib-valid and miniF2F-valid. Performance is reported in", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 544, 505, 557 ], "spans": [ { "bbox": [ 105, 544, 505, 557 ], "score": 1.0, "content": "terms of pass rate (percentage of successful proof searches) as a function of the number of attempts", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 555, 505, 567 ], "spans": [ { "bbox": [ 105, 555, 207, 567 ], "score": 1.0, "content": "per statement, noted pass", "type": "text" }, { "bbox": [ 208, 556, 221, 566 ], "score": 0.56, "content": "@ k", "type": "inline_equation" }, { "bbox": [ 222, 555, 248, 567 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 249, 556, 256, 565 ], "score": 0.81, "content": "k", "type": "inline_equation" }, { "bbox": [ 256, 555, 505, 567 ], "score": 1.0, "content": "is the number of attempts per statement at test time. To reduce", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 566, 505, 578 ], "spans": [ { "bbox": [ 105, 566, 264, 578 ], "score": 1.0, "content": "noise in these metrics we run more than", "type": "text" }, { "bbox": [ 264, 567, 271, 576 ], "score": 0.79, "content": "k", "type": "inline_equation" }, { "bbox": [ 272, 566, 505, 578 ], "score": 1.0, "content": "attempts at test time (generally 32 to compute pass@1 and", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 577, 438, 590 ], "spans": [ { "bbox": [ 106, 577, 142, 589 ], "score": 0.28, "content": "p a s s @ 8 )", "type": "inline_equation" }, { "bbox": [ 142, 577, 397, 590 ], "score": 1.0, "content": "), averaging across attempts as needed to obtain a smoother pass", "type": "text" }, { "bbox": [ 397, 578, 411, 587 ], "score": 0.68, "content": "@ k", "type": "inline_equation" }, { "bbox": [ 411, 577, 438, 590 ], "score": 1.0, "content": "value.", "type": "text" } ], "index": 38 } ], "index": 35.5 }, { "type": "text", "bbox": [ 107, 594, 505, 671 ], "lines": [ { "bbox": [ 106, 594, 504, 606 ], "spans": [ { "bbox": [ 106, 594, 430, 606 ], "score": 1.0, "content": "Given the large number of statements in mathlib-train (25k) we uniformly set", "type": "text" }, { "bbox": [ 430, 595, 457, 604 ], "score": 0.89, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 458, 594, 493, 606 ], "score": 1.0, "content": "and use", "type": "text" }, { "bbox": [ 493, 594, 504, 605 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" } ], "index": 39 }, { "bbox": [ 105, 605, 507, 617 ], "spans": [ { "bbox": [ 105, 605, 123, 617 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 605, 135, 616 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 135, 605, 311, 617 ], "score": 1.0, "content": "as described in Section 4.3 and report pass", "type": "text" }, { "bbox": [ 312, 605, 326, 615 ], "score": 0.56, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 326, 605, 364, 617 ], "score": 1.0, "content": "and pass", "type": "text" }, { "bbox": [ 365, 605, 379, 615 ], "score": 0.51, "content": "@ 8", "type": "inline_equation" }, { "bbox": [ 380, 605, 507, 617 ], "score": 1.0, "content": "across 8 iterations in Figure 1.", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 615, 506, 629 ], "spans": [ { "bbox": [ 105, 615, 144, 629 ], "score": 1.0, "content": "The pass", "type": "text" }, { "bbox": [ 144, 617, 158, 627 ], "score": 0.67, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 159, 615, 274, 629 ], "score": 1.0, "content": "on mathlib-valid goes from", "type": "text" }, { "bbox": [ 274, 616, 302, 627 ], "score": 0.88, "content": "5 6 . 3 \\%", "type": "inline_equation" }, { "bbox": [ 302, 615, 317, 629 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 317, 616, 328, 627 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 328, 615, 340, 629 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 340, 616, 367, 627 ], "score": 0.88, "content": "6 2 . 6 \\%", "type": "inline_equation" }, { "bbox": [ 368, 615, 383, 629 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 383, 616, 393, 627 ], "score": 0.86, "content": "\\theta _ { 9 }", "type": "inline_equation" }, { "bbox": [ 394, 615, 506, 629 ], "score": 1.0, "content": ". The performance steadily", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 627, 506, 640 ], "spans": [ { "bbox": [ 106, 627, 506, 640 ], "score": 1.0, "content": "improves and follows a clear logarithmic scaling law on mathlib-valid. It is also notable that, initially,", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 638, 505, 650 ], "spans": [ { "bbox": [ 105, 638, 505, 650 ], "score": 1.0, "content": "transfer to out-of-distribution miniF2F-valid appears limited but eventually kicks in as we reach", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 649, 505, 661 ], "spans": [ { "bbox": [ 106, 649, 505, 661 ], "score": 1.0, "content": "better performance on mathlib-valid. This demonstrates that the expert iteration process does not just", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 659, 473, 673 ], "spans": [ { "bbox": [ 105, 659, 473, 673 ], "score": 1.0, "content": "overfit to mathlib but also leads to improved performance on out-of-distribution statements.", "type": "text" } ], "index": 45 } ], "index": 42 }, { "type": "text", "bbox": [ 107, 677, 505, 732 ], "lines": [ { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 299, 690 ], "score": 1.0, "content": "We define the cumulative pass rate at iteration", "type": "text" }, { "bbox": [ 300, 677, 307, 687 ], "score": 0.81, "content": "k", "type": "inline_equation" }, { "bbox": [ 307, 676, 505, 690 ], "score": 1.0, "content": "as the pass rate consisting of all proof searches", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 688, 505, 700 ], "spans": [ { "bbox": [ 105, 688, 166, 700 ], "score": 1.0, "content": "up to iteration", "type": "text" }, { "bbox": [ 166, 688, 173, 698 ], "score": 0.65, "content": "k", "type": "inline_equation" }, { "bbox": [ 174, 688, 234, 700 ], "score": 1.0, "content": ". Since we set", "type": "text" }, { "bbox": [ 234, 688, 264, 698 ], "score": 0.89, "content": "a = 1 6", "type": "inline_equation" }, { "bbox": [ 264, 688, 436, 700 ], "score": 1.0, "content": "for evaluation on mathlib-valid and miniF", "type": "text" }, { "bbox": [ 436, 688, 449, 698 ], "score": 0.51, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 449, 688, 505, 700 ], "score": 1.0, "content": "-valid at each", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 698, 505, 712 ], "spans": [ { "bbox": [ 106, 698, 284, 712 ], "score": 1.0, "content": "iteration, the cumulative pass rate at iteration", "type": "text" }, { "bbox": [ 284, 699, 291, 709 ], "score": 0.81, "content": "k", "type": "inline_equation" }, { "bbox": [ 291, 698, 505, 712 ], "score": 1.0, "content": "can be seen as a noisy ensembled pass@16k (multiple", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 708, 506, 723 ], "spans": [ { "bbox": [ 105, 708, 138, 723 ], "score": 1.0, "content": "models", "type": "text" }, { "bbox": [ 138, 710, 155, 721 ], "score": 0.84, "content": "( \\theta _ { k } )", "type": "inline_equation" }, { "bbox": [ 155, 708, 506, 723 ], "score": 1.0, "content": ", no averaging). In Figure 2, we report this cumulative pass rate for two iteration loops,", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "our normal one and a sampling-only loop where we skip re-training the model between iterations", "type": "text" } ], "index": 50 } ], "index": 48 } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 763 ], "spans": [ { "bbox": [ 302, 750, 309, 763 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 137 ], "lines": [ { "bbox": [ 106, 83, 505, 95 ], "spans": [ { "bbox": [ 106, 83, 229, 95 ], "score": 1.0, "content": "Interestingly, the gap between", "type": "text" }, { "bbox": [ 230, 83, 240, 93 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 241, 83, 258, 95 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 258, 83, 269, 93 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 269, 83, 306, 95 ], "score": 1.0, "content": "on miniF", "type": "text" }, { "bbox": [ 306, 83, 320, 93 ], "score": 0.28, "content": "{ } ^ { 7 2 F }", "type": "inline_equation" }, { "bbox": [ 320, 83, 505, 95 ], "score": 1.0, "content": "-valid is not as significant, demonstrating that", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 106 ], "spans": [ { "bbox": [ 105, 93, 506, 106 ], "score": 1.0, "content": "training a value function on proofs sampled from mathlib-train has limited transfer to miniF2F-valid.", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 505, 117 ], "score": 1.0, "content": "The main differences with Zheng et al. (2022), potentially explaining the gap on miniF2F-valid", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 114, 506, 128 ], "spans": [ { "bbox": [ 107, 115, 137, 127 ], "score": 0.84, "content": "( 2 7 . 6 \\%", "type": "inline_equation" }, { "bbox": [ 137, 114, 149, 128 ], "score": 1.0, "content": "vs", "type": "text" }, { "bbox": [ 150, 115, 178, 126 ], "score": 0.86, "content": "2 3 . 9 \\%", "type": "inline_equation" }, { "bbox": [ 178, 114, 506, 128 ], "score": 1.0, "content": "), consists in the new pre-training described in Section 4.1 as well as the use of a", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 127, 397, 139 ], "spans": [ { "bbox": [ 106, 127, 397, 139 ], "score": 1.0, "content": "more recent mathlib checkpoint for the mix1, mix2 and tactic datasets.", "type": "text" } ], "index": 4 } ], "index": 2, "bbox_fs": [ 105, 83, 506, 139 ] }, { "type": "title", "bbox": [ 108, 153, 287, 164 ], "lines": [ { "bbox": [ 105, 153, 289, 165 ], "spans": [ { "bbox": [ 105, 153, 289, 165 ], "score": 1.0, "content": "4.4 ITERATED SAMPLING AND TRAINING", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 107, 174, 505, 218 ], "lines": [ { "bbox": [ 105, 174, 505, 187 ], "spans": [ { "bbox": [ 105, 174, 387, 187 ], "score": 1.0, "content": "Our expert iteration process takes as input: (i) a set of formal statements", "type": "text" }, { "bbox": [ 387, 175, 398, 185 ], "score": 0.82, "content": "S t", "type": "inline_equation" }, { "bbox": [ 398, 174, 456, 187 ], "score": 1.0, "content": ", (ii) a function", "type": "text" }, { "bbox": [ 456, 174, 505, 185 ], "score": 0.9, "content": "a : S t \\mathbb { N }", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 105, 185, 505, 198 ], "spans": [ { "bbox": [ 105, 185, 505, 198 ], "score": 1.0, "content": "indicating the number of proof search attempts to run per statement at each iteration, (iii) a base", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 196, 505, 208 ], "spans": [ { "bbox": [ 106, 197, 133, 208 ], "score": 1.0, "content": "model", "type": "text" }, { "bbox": [ 133, 197, 144, 207 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 144, 197, 432, 208 ], "score": 1.0, "content": "to fine-tune from at each iteration, and (iv) a mathlib bootstrapped model", "type": "text" }, { "bbox": [ 432, 196, 442, 207 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 442, 197, 505, 208 ], "score": 1.0, "content": "trained on both", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 208, 506, 219 ], "spans": [ { "bbox": [ 106, 208, 506, 219 ], "score": 1.0, "content": "objectives. A high-level illustration of the iterated sampling and training is available in Appendix E.", "type": "text" } ], "index": 9 } ], "index": 7.5, "bbox_fs": [ 105, 174, 506, 219 ] }, { "type": "text", "bbox": [ 107, 223, 505, 290 ], "lines": [ { "bbox": [ 106, 224, 505, 236 ], "spans": [ { "bbox": [ 106, 224, 163, 236 ], "score": 1.0, "content": "Each iteration", "type": "text" }, { "bbox": [ 163, 225, 170, 234 ], "score": 0.8, "content": "k", "type": "inline_equation" }, { "bbox": [ 171, 224, 379, 236 ], "score": 1.0, "content": "consists in sampling proof searches for statements in", "type": "text" }, { "bbox": [ 379, 225, 390, 234 ], "score": 0.8, "content": "S t", "type": "inline_equation" }, { "bbox": [ 390, 224, 414, 236 ], "score": 1.0, "content": "using", "type": "text" }, { "bbox": [ 415, 224, 425, 235 ], "score": 0.88, "content": "\\theta _ { k }", "type": "inline_equation" }, { "bbox": [ 425, 224, 505, 236 ], "score": 1.0, "content": ", filtering successful", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 235, 505, 247 ], "spans": [ { "bbox": [ 105, 235, 168, 247 ], "score": 1.0, "content": "proof searches", "type": "text" }, { "bbox": [ 169, 235, 181, 246 ], "score": 0.88, "content": "S _ { k }", "type": "inline_equation" }, { "bbox": [ 181, 235, 282, 247 ], "score": 1.0, "content": "to extract a new dataset", "type": "text" }, { "bbox": [ 282, 235, 297, 246 ], "score": 0.89, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 297, 235, 366, 247 ], "score": 1.0, "content": ", and fine-tuning", "type": "text" }, { "bbox": [ 366, 235, 376, 246 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 377, 235, 439, 247 ], "score": 1.0, "content": "on it to obtain", "type": "text" }, { "bbox": [ 439, 235, 460, 247 ], "score": 0.91, "content": "\\theta _ { k + 1 }", "type": "inline_equation" }, { "bbox": [ 460, 235, 505, 247 ], "score": 1.0, "content": ", on which", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 246, 505, 258 ], "spans": [ { "bbox": [ 105, 246, 297, 258 ], "score": 1.0, "content": "we can iterate. To sample proof searches from", "type": "text" }, { "bbox": [ 297, 246, 309, 256 ], "score": 0.81, "content": "S t", "type": "inline_equation" }, { "bbox": [ 309, 246, 495, 258 ], "score": 1.0, "content": "we use the best-first search described in Polu", "type": "text" }, { "bbox": [ 496, 246, 505, 256 ], "score": 0.27, "content": "\\&", "type": "inline_equation" } ], "index": 12 }, { "bbox": [ 105, 257, 505, 269 ], "spans": [ { "bbox": [ 105, 257, 422, 269 ], "score": 1.0, "content": "Sutskever (2020) with the value function described in Section 4.2. We attempt", "type": "text" }, { "bbox": [ 423, 259, 429, 267 ], "score": 0.66, "content": "a", "type": "inline_equation" }, { "bbox": [ 429, 257, 505, 269 ], "score": 1.0, "content": "proof searches for", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 268, 505, 281 ], "spans": [ { "bbox": [ 105, 268, 167, 281 ], "score": 1.0, "content": "each statement", "type": "text" }, { "bbox": [ 167, 268, 207, 280 ], "score": 0.92, "content": "s ( s \\in S t )", "type": "inline_equation" }, { "bbox": [ 208, 268, 228, 281 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 229, 268, 264, 278 ], "score": 0.9, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 264, 268, 327, 281 ], "score": 1.0, "content": "expansions and", "type": "text" }, { "bbox": [ 327, 268, 352, 278 ], "score": 0.9, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 352, 268, 505, 281 ], "score": 1.0, "content": "samples per expansion. We denote the", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 279, 319, 291 ], "spans": [ { "bbox": [ 105, 279, 284, 291 ], "score": 1.0, "content": "set of successful proof searches for iteration", "type": "text" }, { "bbox": [ 284, 280, 291, 289 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 292, 279, 303, 291 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 303, 279, 315, 290 ], "score": 0.89, "content": "S _ { k }", "type": "inline_equation" }, { "bbox": [ 315, 279, 319, 291 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 12.5, "bbox_fs": [ 105, 224, 505, 291 ] }, { "type": "text", "bbox": [ 106, 295, 505, 343 ], "lines": [ { "bbox": [ 106, 295, 506, 309 ], "spans": [ { "bbox": [ 106, 295, 133, 309 ], "score": 1.0, "content": "Using", "type": "text" }, { "bbox": [ 134, 296, 146, 307 ], "score": 0.87, "content": "S _ { k }", "type": "inline_equation" }, { "bbox": [ 146, 295, 235, 309 ], "score": 1.0, "content": "we generate datasets", "type": "text" }, { "bbox": [ 235, 297, 249, 307 ], "score": 0.92, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 250, 295, 506, 309 ], "score": 1.0, "content": "by concatenating: (i) the initial tactic dataset (proofstep ob-", "type": "text" } ], "index": 16 }, { "bbox": [ 104, 304, 506, 322 ], "spans": [ { "bbox": [ 104, 304, 412, 322 ], "score": 1.0, "content": "jective), (ii) a deduplicated set of proofsteps extracted from the proofs in", "type": "text" }, { "bbox": [ 412, 307, 459, 320 ], "score": 0.92, "content": "\\textstyle \\bigcup _ { 1 \\leq i \\leq k } { \\bar { S } } _ { k }", "type": "inline_equation" }, { "bbox": [ 460, 304, 506, 322 ], "score": 1.0, "content": "(proofstep", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 318, 505, 332 ], "spans": [ { "bbox": [ 105, 318, 505, 332 ], "score": 1.0, "content": "objective), and (iii) a deduplicated set of proofsize tuples (goals and proofsize) extracted from the", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 329, 328, 344 ], "spans": [ { "bbox": [ 104, 329, 193, 344 ], "score": 1.0, "content": "full proof searches in", "type": "text" }, { "bbox": [ 193, 330, 240, 344 ], "score": 0.93, "content": "\\textstyle \\bigcup _ { 1 \\leq i \\leq k } S _ { k }", "type": "inline_equation" }, { "bbox": [ 240, 329, 328, 344 ], "score": 1.0, "content": "(proofsize objective).", "type": "text" } ], "index": 19 } ], "index": 17.5, "bbox_fs": [ 104, 295, 506, 344 ] }, { "type": "text", "bbox": [ 107, 347, 505, 425 ], "lines": [ { "bbox": [ 105, 347, 505, 360 ], "spans": [ { "bbox": [ 105, 347, 505, 360 ], "score": 1.0, "content": "We use a global deduplication across iterations for both proofsteps and proofsize tuples which", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 358, 506, 371 ], "spans": [ { "bbox": [ 105, 358, 506, 371 ], "score": 1.0, "content": "we found to be important to maintain the stability of the expert iteration procedure. This global", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 369, 505, 383 ], "spans": [ { "bbox": [ 105, 369, 505, 383 ], "score": 1.0, "content": "deduplication is somewhat equivalent for each statement to growing a unique proof tree by aggregating", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 381, 505, 392 ], "spans": [ { "bbox": [ 106, 381, 505, 392 ], "score": 1.0, "content": "all the proof searches that have been run for it across iterations. This virtual proof tree accumulates a", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 392, 506, 405 ], "spans": [ { "bbox": [ 105, 392, 506, 405 ], "score": 1.0, "content": "growing number of positive proof paths and visited goals that remain unproven. We use these goals", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 403, 505, 415 ], "spans": [ { "bbox": [ 105, 403, 505, 415 ], "score": 1.0, "content": "as negative examples for the proofsize objective, labeling them with an infinite proofsize. Positive", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 414, 424, 426 ], "spans": [ { "bbox": [ 106, 414, 424, 426 ], "score": 1.0, "content": "goals are deduplicated keeping the minimum proof sizes across proof searches.", "type": "text" } ], "index": 26 } ], "index": 23, "bbox_fs": [ 105, 347, 506, 426 ] }, { "type": "text", "bbox": [ 107, 430, 505, 485 ], "lines": [ { "bbox": [ 105, 429, 506, 443 ], "spans": [ { "bbox": [ 105, 429, 138, 443 ], "score": 1.0, "content": "Finally", "type": "text" }, { "bbox": [ 138, 431, 149, 442 ], "score": 0.88, "content": "\\theta _ { k }", "type": "inline_equation" }, { "bbox": [ 149, 429, 257, 443 ], "score": 1.0, "content": "is obtained by fine-tuning", "type": "text" }, { "bbox": [ 257, 431, 267, 442 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 268, 429, 372, 443 ], "score": 1.0, "content": "for exactly one epoch on", "type": "text" }, { "bbox": [ 372, 431, 386, 442 ], "score": 0.89, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 386, 429, 506, 443 ], "score": 1.0, "content": ". Note that the initial tactic", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 440, 505, 454 ], "spans": [ { "bbox": [ 105, 440, 215, 454 ], "score": 1.0, "content": "dataset is included in each", "type": "text" }, { "bbox": [ 215, 442, 229, 452 ], "score": 0.88, "content": "D _ { k }", "type": "inline_equation" }, { "bbox": [ 230, 440, 264, 454 ], "score": 1.0, "content": ", despite", "type": "text" }, { "bbox": [ 265, 442, 275, 453 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 275, 440, 505, 454 ], "score": 1.0, "content": "being already trained on it (along with mix1). We found", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 453, 505, 464 ], "spans": [ { "bbox": [ 106, 453, 505, 464 ], "score": 1.0, "content": "this repetition to be beneficial overall (as it adds the mathlib extracted proofsteps to our deduplicated", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 463, 506, 476 ], "spans": [ { "bbox": [ 105, 463, 506, 476 ], "score": 1.0, "content": "per statements virtual proof trees) despite it leading to a slight overfit on the tactic dataset in terms", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 473, 180, 486 ], "spans": [ { "bbox": [ 105, 473, 180, 486 ], "score": 1.0, "content": "of validation loss.", "type": "text" } ], "index": 31 } ], "index": 29, "bbox_fs": [ 105, 429, 506, 486 ] }, { "type": "title", "bbox": [ 109, 501, 284, 512 ], "lines": [ { "bbox": [ 106, 500, 286, 514 ], "spans": [ { "bbox": [ 106, 500, 286, 514 ], "score": 1.0, "content": "4.5 EXPERT ITERATION ON mathlib-train", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 107, 522, 505, 588 ], "lines": [ { "bbox": [ 105, 522, 505, 535 ], "spans": [ { "bbox": [ 105, 522, 234, 535 ], "score": 1.0, "content": "In this section we propose to set", "type": "text" }, { "bbox": [ 234, 523, 245, 532 ], "score": 0.83, "content": "S t", "type": "inline_equation" }, { "bbox": [ 245, 522, 505, 535 ], "score": 1.0, "content": "to the statements in mathlib-train, run our expert iteration process", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 533, 505, 545 ], "spans": [ { "bbox": [ 106, 533, 505, 545 ], "score": 1.0, "content": "with it and report performance on both mathlib-valid and miniF2F-valid. Performance is reported in", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 544, 505, 557 ], "spans": [ { "bbox": [ 105, 544, 505, 557 ], "score": 1.0, "content": "terms of pass rate (percentage of successful proof searches) as a function of the number of attempts", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 555, 505, 567 ], "spans": [ { "bbox": [ 105, 555, 207, 567 ], "score": 1.0, "content": "per statement, noted pass", "type": "text" }, { "bbox": [ 208, 556, 221, 566 ], "score": 0.56, "content": "@ k", "type": "inline_equation" }, { "bbox": [ 222, 555, 248, 567 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 249, 556, 256, 565 ], "score": 0.81, "content": "k", "type": "inline_equation" }, { "bbox": [ 256, 555, 505, 567 ], "score": 1.0, "content": "is the number of attempts per statement at test time. To reduce", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 566, 505, 578 ], "spans": [ { "bbox": [ 105, 566, 264, 578 ], "score": 1.0, "content": "noise in these metrics we run more than", "type": "text" }, { "bbox": [ 264, 567, 271, 576 ], "score": 0.79, "content": "k", "type": "inline_equation" }, { "bbox": [ 272, 566, 505, 578 ], "score": 1.0, "content": "attempts at test time (generally 32 to compute pass@1 and", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 577, 438, 590 ], "spans": [ { "bbox": [ 106, 577, 142, 589 ], "score": 0.28, "content": "p a s s @ 8 )", "type": "inline_equation" }, { "bbox": [ 142, 577, 397, 590 ], "score": 1.0, "content": "), averaging across attempts as needed to obtain a smoother pass", "type": "text" }, { "bbox": [ 397, 578, 411, 587 ], "score": 0.68, "content": "@ k", "type": "inline_equation" }, { "bbox": [ 411, 577, 438, 590 ], "score": 1.0, "content": "value.", "type": "text" } ], "index": 38 } ], "index": 35.5, "bbox_fs": [ 105, 522, 505, 590 ] }, { "type": "text", "bbox": [ 107, 594, 505, 671 ], "lines": [ { "bbox": [ 106, 594, 504, 606 ], "spans": [ { "bbox": [ 106, 594, 430, 606 ], "score": 1.0, "content": "Given the large number of statements in mathlib-train (25k) we uniformly set", "type": "text" }, { "bbox": [ 430, 595, 457, 604 ], "score": 0.89, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 458, 594, 493, 606 ], "score": 1.0, "content": "and use", "type": "text" }, { "bbox": [ 493, 594, 504, 605 ], "score": 0.87, "content": "\\theta _ { 0 }", "type": "inline_equation" } ], "index": 39 }, { "bbox": [ 105, 605, 507, 617 ], "spans": [ { "bbox": [ 105, 605, 123, 617 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 605, 135, 616 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 135, 605, 311, 617 ], "score": 1.0, "content": "as described in Section 4.3 and report pass", "type": "text" }, { "bbox": [ 312, 605, 326, 615 ], "score": 0.56, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 326, 605, 364, 617 ], "score": 1.0, "content": "and pass", "type": "text" }, { "bbox": [ 365, 605, 379, 615 ], "score": 0.51, "content": "@ 8", "type": "inline_equation" }, { "bbox": [ 380, 605, 507, 617 ], "score": 1.0, "content": "across 8 iterations in Figure 1.", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 615, 506, 629 ], "spans": [ { "bbox": [ 105, 615, 144, 629 ], "score": 1.0, "content": "The pass", "type": "text" }, { "bbox": [ 144, 617, 158, 627 ], "score": 0.67, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 159, 615, 274, 629 ], "score": 1.0, "content": "on mathlib-valid goes from", "type": "text" }, { "bbox": [ 274, 616, 302, 627 ], "score": 0.88, "content": "5 6 . 3 \\%", "type": "inline_equation" }, { "bbox": [ 302, 615, 317, 629 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 317, 616, 328, 627 ], "score": 0.87, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 328, 615, 340, 629 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 340, 616, 367, 627 ], "score": 0.88, "content": "6 2 . 6 \\%", "type": "inline_equation" }, { "bbox": [ 368, 615, 383, 629 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 383, 616, 393, 627 ], "score": 0.86, "content": "\\theta _ { 9 }", "type": "inline_equation" }, { "bbox": [ 394, 615, 506, 629 ], "score": 1.0, "content": ". The performance steadily", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 627, 506, 640 ], "spans": [ { "bbox": [ 106, 627, 506, 640 ], "score": 1.0, "content": "improves and follows a clear logarithmic scaling law on mathlib-valid. It is also notable that, initially,", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 638, 505, 650 ], "spans": [ { "bbox": [ 105, 638, 505, 650 ], "score": 1.0, "content": "transfer to out-of-distribution miniF2F-valid appears limited but eventually kicks in as we reach", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 649, 505, 661 ], "spans": [ { "bbox": [ 106, 649, 505, 661 ], "score": 1.0, "content": "better performance on mathlib-valid. This demonstrates that the expert iteration process does not just", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 659, 473, 673 ], "spans": [ { "bbox": [ 105, 659, 473, 673 ], "score": 1.0, "content": "overfit to mathlib but also leads to improved performance on out-of-distribution statements.", "type": "text" } ], "index": 45 } ], "index": 42, "bbox_fs": [ 105, 594, 507, 673 ] }, { "type": "text", "bbox": [ 107, 677, 505, 732 ], "lines": [ { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 299, 690 ], "score": 1.0, "content": "We define the cumulative pass rate at iteration", "type": "text" }, { "bbox": [ 300, 677, 307, 687 ], "score": 0.81, "content": "k", "type": "inline_equation" }, { "bbox": [ 307, 676, 505, 690 ], "score": 1.0, "content": "as the pass rate consisting of all proof searches", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 688, 505, 700 ], "spans": [ { "bbox": [ 105, 688, 166, 700 ], "score": 1.0, "content": "up to iteration", "type": "text" }, { "bbox": [ 166, 688, 173, 698 ], "score": 0.65, "content": "k", "type": "inline_equation" }, { "bbox": [ 174, 688, 234, 700 ], "score": 1.0, "content": ". Since we set", "type": "text" }, { "bbox": [ 234, 688, 264, 698 ], "score": 0.89, "content": "a = 1 6", "type": "inline_equation" }, { "bbox": [ 264, 688, 436, 700 ], "score": 1.0, "content": "for evaluation on mathlib-valid and miniF", "type": "text" }, { "bbox": [ 436, 688, 449, 698 ], "score": 0.51, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 449, 688, 505, 700 ], "score": 1.0, "content": "-valid at each", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 698, 505, 712 ], "spans": [ { "bbox": [ 106, 698, 284, 712 ], "score": 1.0, "content": "iteration, the cumulative pass rate at iteration", "type": "text" }, { "bbox": [ 284, 699, 291, 709 ], "score": 0.81, "content": "k", "type": "inline_equation" }, { "bbox": [ 291, 698, 505, 712 ], "score": 1.0, "content": "can be seen as a noisy ensembled pass@16k (multiple", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 708, 506, 723 ], "spans": [ { "bbox": [ 105, 708, 138, 723 ], "score": 1.0, "content": "models", "type": "text" }, { "bbox": [ 138, 710, 155, 721 ], "score": 0.84, "content": "( \\theta _ { k } )", "type": "inline_equation" }, { "bbox": [ 155, 708, 506, 723 ], "score": 1.0, "content": ", no averaging). In Figure 2, we report this cumulative pass rate for two iteration loops,", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "our normal one and a sampling-only loop where we skip re-training the model between iterations", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 349, 505, 362 ], "spans": [ { "bbox": [ 105, 349, 203, 362 ], "score": 1.0, "content": "and solely sample from", "type": "text", "cross_page": true }, { "bbox": [ 203, 350, 213, 361 ], "score": 0.86, "content": "\\theta _ { 1 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 213, 349, 505, 362 ], "score": 1.0, "content": ". This directly compares test-time compute scaling (scaling proof search", "type": "text", "cross_page": true } ], "index": 39 }, { "bbox": [ 105, 361, 506, 373 ], "spans": [ { "bbox": [ 105, 361, 506, 373 ], "score": 1.0, "content": "attempts) to expert iteration scaling (interleaved training on new data sampled from mathlib-train)", "type": "text", "cross_page": true } ], "index": 40 }, { "bbox": [ 106, 372, 505, 384 ], "spans": [ { "bbox": [ 106, 372, 505, 384 ], "score": 1.0, "content": "and provides a very clear visualization of the gains of expert iteration. For a fair comparison, we", "type": "text", "cross_page": true } ], "index": 41 }, { "bbox": [ 105, 383, 506, 395 ], "spans": [ { "bbox": [ 105, 383, 506, 395 ], "score": 1.0, "content": "also report an adjusted compute line which approximates the test-time performance we would get", "type": "text", "cross_page": true } ], "index": 42 }, { "bbox": [ 105, 392, 505, 407 ], "spans": [ { "bbox": [ 105, 392, 505, 407 ], "score": 1.0, "content": "at each iteration if we were to focus all the additional compute used by expert iteration (sampling", "type": "text", "cross_page": true } ], "index": 43 }, { "bbox": [ 105, 404, 505, 418 ], "spans": [ { "bbox": [ 105, 404, 505, 418 ], "score": 1.0, "content": "proofs from mathlib-train as well as re-training models at each iteration) towards solely running", "type": "text", "cross_page": true } ], "index": 44 }, { "bbox": [ 105, 416, 256, 427 ], "spans": [ { "bbox": [ 105, 416, 256, 427 ], "score": 1.0, "content": "proof searches against mathlib-valid.", "type": "text", "cross_page": true } ], "index": 45 } ], "index": 48, "bbox_fs": [ 105, 676, 506, 734 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 307, 80, 497, 199 ], "blocks": [ { "type": "image_caption", "bbox": [ 114, 211, 302, 321 ], "group_id": 1, "lines": [ { "bbox": [ 113, 211, 304, 223 ], "spans": [ { "bbox": [ 113, 211, 254, 223 ], "score": 1.0, "content": "Figure 1: pass@1 (plain) and pass", "type": "text" }, { "bbox": [ 254, 212, 269, 222 ], "score": 0.66, "content": "@ 8", "type": "inline_equation" }, { "bbox": [ 269, 211, 304, 223 ], "score": 1.0, "content": "(dotted)", "type": "text" } ], "index": 9 }, { "bbox": [ 114, 222, 304, 233 ], "spans": [ { "bbox": [ 114, 222, 225, 233 ], "score": 1.0, "content": "for mathlib-valid and miniF", "type": "text" }, { "bbox": [ 225, 222, 237, 232 ], "score": 0.52, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 238, 222, 304, 233 ], "score": 1.0, "content": "-valid when run-", "type": "text" } ], "index": 10 }, { "bbox": [ 113, 233, 303, 244 ], "spans": [ { "bbox": [ 113, 233, 236, 244 ], "score": 1.0, "content": "ning 8 expert iterations with", "type": "text" }, { "bbox": [ 236, 233, 247, 243 ], "score": 0.77, "content": "S t", "type": "inline_equation" }, { "bbox": [ 248, 233, 303, 244 ], "score": 1.0, "content": "set to be the", "type": "text" } ], "index": 11 }, { "bbox": [ 113, 243, 304, 256 ], "spans": [ { "bbox": [ 113, 243, 248, 256 ], "score": 1.0, "content": "statements in mathlib-train. 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This directly compares test-time compute scaling (scaling proof search", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 361, 506, 373 ], "spans": [ { "bbox": [ 105, 361, 506, 373 ], "score": 1.0, "content": "attempts) to expert iteration scaling (interleaved training on new data sampled from mathlib-train)", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 372, 505, 384 ], "spans": [ { "bbox": [ 106, 372, 505, 384 ], "score": 1.0, "content": "and provides a very clear visualization of the gains of expert iteration. For a fair comparison, we", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 383, 506, 395 ], "spans": [ { "bbox": [ 105, 383, 506, 395 ], "score": 1.0, "content": "also report an adjusted compute line which approximates the test-time performance we would get", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 392, 505, 407 ], "spans": [ { "bbox": [ 105, 392, 505, 407 ], "score": 1.0, "content": "at each iteration if we were to focus all the additional compute used by expert iteration (sampling", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 404, 505, 418 ], "spans": [ { "bbox": [ 105, 404, 505, 418 ], "score": 1.0, "content": "proofs from mathlib-train as well as re-training models at each iteration) towards solely running", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 416, 256, 427 ], "spans": [ { "bbox": [ 105, 416, 256, 427 ], "score": 1.0, "content": "proof searches against mathlib-valid.", "type": "text" } ], "index": 45 } ], "index": 42 }, { "type": "text", "bbox": [ 107, 432, 505, 477 ], "lines": [ { "bbox": [ 105, 432, 505, 445 ], "spans": [ { "bbox": [ 105, 432, 505, 445 ], "score": 1.0, "content": "As shown by Figure 2, the scaling exponent of expert iteration is substantially higher than the", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 443, 506, 456 ], "spans": [ { "bbox": [ 105, 443, 506, 456 ], "score": 1.0, "content": "scaling exponent associated with solely scaling test-time compute (running more proof searches),", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 454, 505, 467 ], "spans": [ { "bbox": [ 105, 454, 505, 467 ], "score": 1.0, "content": "demonstrating the clear benefit of expert iteration. We’ll denote the fully iterated model from this", "type": "text" } ], "index": 48 }, { "bbox": [ 103, 463, 181, 479 ], "spans": [ { "bbox": [ 103, 463, 181, 479 ], "score": 1.0, "content": "section as θmathlib9 .", "type": "text" } ], "index": 49 } ], "index": 47.5 }, { "type": "text", "bbox": [ 107, 482, 505, 592 ], "lines": [ { "bbox": [ 105, 481, 506, 495 ], "spans": [ { "bbox": [ 105, 481, 506, 495 ], "score": 1.0, "content": "Even in the presence of ground-truth proofs for each of the statements in mathlib-train (tactic", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 493, 506, 505 ], "spans": [ { "bbox": [ 105, 493, 506, 505 ], "score": 1.0, "content": "dataset), expert iteration generates data that further improves the performance of the model. The", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 504, 506, 517 ], "spans": [ { "bbox": [ 105, 504, 374, 517 ], "score": 1.0, "content": "number of statements proved in mathlib-train goes from 17390", "type": "text" }, { "bbox": [ 374, 504, 407, 515 ], "score": 0.77, "content": "( 6 7 . 8 \\% )", "type": "inline_equation" }, { "bbox": [ 407, 504, 506, 517 ], "score": 1.0, "content": "at iteration 1 to 19476", "type": "text" } ], "index": 52 }, { "bbox": [ 106, 514, 506, 528 ], "spans": [ { "bbox": [ 106, 515, 140, 526 ], "score": 0.86, "content": "( 7 6 . 0 \\% )", "type": "inline_equation" }, { "bbox": [ 140, 514, 506, 528 ], "score": 1.0, "content": "at iteration 9, while the average proof length of these statements goes from 4.8 to 4.0. We", "type": "text" } ], "index": 53 }, { "bbox": [ 105, 526, 506, 539 ], "spans": [ { "bbox": [ 105, 526, 506, 539 ], "score": 1.0, "content": "hypothesize that this continuously improving performance through expert iteration stems from two", "type": "text" } ], "index": 54 }, { "bbox": [ 105, 536, 506, 551 ], "spans": [ { "bbox": [ 105, 536, 506, 551 ], "score": 1.0, "content": "effects: (i) the model finding new original proofs for the same statements and (ii) the model closing", "type": "text" } ], "index": 55 }, { "bbox": [ 105, 549, 505, 560 ], "spans": [ { "bbox": [ 105, 549, 505, 560 ], "score": 1.0, "content": "marginally harder statements at each iteration – which in turn provides more useful training data for", "type": "text" } ], "index": 56 }, { "bbox": [ 105, 558, 506, 572 ], "spans": [ { "bbox": [ 105, 558, 374, 572 ], "score": 1.0, "content": "the next iteration. By iteration 9, the model is trained on more than", "type": "text" }, { "bbox": [ 374, 559, 394, 570 ], "score": 0.88, "content": "9 0 \\%", "type": "inline_equation" }, { "bbox": [ 395, 558, 506, 572 ], "score": 1.0, "content": "generated data. We present", "type": "text" } ], "index": 57 }, { "bbox": [ 105, 570, 505, 582 ], "spans": [ { "bbox": [ 105, 570, 505, 582 ], "score": 1.0, "content": "in Appendix I a few examples of original proofs found by our models on mathlib-train compared", "type": "text" } ], "index": 58 }, { "bbox": [ 105, 581, 239, 594 ], "spans": [ { "bbox": [ 105, 581, 239, 594 ], "score": 1.0, "content": "with their ground-truth versions.", "type": "text" } ], "index": 59 } ], "index": 54.5 }, { "type": "text", "bbox": [ 107, 597, 505, 653 ], "lines": [ { "bbox": [ 105, 597, 505, 610 ], "spans": [ { "bbox": [ 105, 597, 505, 610 ], "score": 1.0, "content": "To verify our hypothesis that expert iteration is capable of closing a curriculum of increasingly", "type": "text" } ], "index": 60 }, { "bbox": [ 105, 606, 506, 623 ], "spans": [ { "bbox": [ 105, 606, 506, 623 ], "score": 1.0, "content": "difficult problems out of a set of problem statements, and that this capability is independent of having", "type": "text" } ], "index": 61 }, { "bbox": [ 105, 619, 506, 632 ], "spans": [ { "bbox": [ 105, 619, 506, 632 ], "score": 1.0, "content": "access to ground-truth proofs, we propose in the next section to study expert iteration applied to a", "type": "text" } ], "index": 62 }, { "bbox": [ 105, 631, 506, 643 ], "spans": [ { "bbox": [ 105, 631, 506, 643 ], "score": 1.0, "content": "synthetically generated set of problems for which we have fine-grained control on the difficulty of", "type": "text" } ], "index": 63 }, { "bbox": [ 106, 642, 170, 654 ], "spans": [ { "bbox": [ 106, 642, 170, 654 ], "score": 1.0, "content": "each statement.", "type": "text" } ], "index": 64 } ], "index": 62 }, { "type": "title", "bbox": [ 108, 672, 317, 684 ], "lines": [ { "bbox": [ 105, 671, 318, 687 ], "spans": [ { "bbox": [ 105, 671, 318, 687 ], "score": 1.0, "content": "5 STATEMENT CURRICULUM LEARNING", "type": "text" } ], "index": 65 } ], "index": 65 }, { "type": "text", "bbox": [ 108, 699, 505, 732 ], "lines": [ { "bbox": [ 104, 698, 505, 712 ], "spans": [ { "bbox": [ 104, 698, 505, 712 ], "score": 1.0, "content": "In this section we focus on running expert iteration on synthetic statements generated by an inequality", "type": "text" } ], "index": 66 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "generator. 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The adjusted compute line is computed", "type": "text" } ], "index": 31 }, { "bbox": [ 308, 248, 497, 261 ], "spans": [ { "bbox": [ 308, 248, 497, 261 ], "score": 1.0, "content": "by fitting the sample only curve and shifting it", "type": "text" } ], "index": 32 }, { "bbox": [ 307, 259, 497, 271 ], "spans": [ { "bbox": [ 307, 259, 497, 271 ], "score": 1.0, "content": "to approximate a setup where we would focus", "type": "text" } ], "index": 33 }, { "bbox": [ 307, 269, 498, 282 ], "spans": [ { "bbox": [ 307, 269, 498, 282 ], "score": 1.0, "content": "all the additional compute used by expert itera-", "type": "text" } ], "index": 34 }, { "bbox": [ 307, 281, 497, 293 ], "spans": [ { "bbox": [ 307, 281, 497, 293 ], "score": 1.0, "content": "tion (sampling training data from mathlib-train", "type": "text" } ], "index": 35 }, { "bbox": [ 307, 292, 498, 304 ], "spans": [ { "bbox": [ 307, 292, 498, 304 ], "score": 1.0, "content": "as well as re-training models at each iteration)", "type": "text" } ], "index": 36 }, { "bbox": [ 307, 303, 498, 315 ], "spans": [ { "bbox": [ 307, 303, 498, 315 ], "score": 1.0, "content": "towards running proof searches against mathlib-", "type": "text" } ], "index": 37 }, { "bbox": [ 307, 313, 334, 326 ], "spans": [ { "bbox": [ 307, 313, 334, 326 ], "score": 1.0, "content": "valid.", "type": "text" } ], "index": 38 } ], "index": 33 } ], "index": 18.5 }, { "type": "text", "bbox": [ 107, 349, 505, 426 ], "lines": [], "index": 42, "bbox_fs": [ 105, 349, 506, 427 ], "lines_deleted": true }, { "type": "text", "bbox": [ 107, 432, 505, 477 ], "lines": [ { "bbox": [ 105, 432, 505, 445 ], "spans": [ { "bbox": [ 105, 432, 505, 445 ], "score": 1.0, "content": "As shown by Figure 2, the scaling exponent of expert iteration is substantially higher than the", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 443, 506, 456 ], "spans": [ { "bbox": [ 105, 443, 506, 456 ], "score": 1.0, "content": "scaling exponent associated with solely scaling test-time compute (running more proof searches),", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 454, 505, 467 ], "spans": [ { "bbox": [ 105, 454, 505, 467 ], "score": 1.0, "content": "demonstrating the clear benefit of expert iteration. We’ll denote the fully iterated model from this", "type": "text" } ], "index": 48 }, { "bbox": [ 103, 463, 181, 479 ], "spans": [ { "bbox": [ 103, 463, 181, 479 ], "score": 1.0, "content": "section as θmathlib9 .", "type": "text" } ], "index": 49 } ], "index": 47.5, "bbox_fs": [ 103, 432, 506, 479 ] }, { "type": "text", "bbox": [ 107, 482, 505, 592 ], "lines": [ { "bbox": [ 105, 481, 506, 495 ], "spans": [ { "bbox": [ 105, 481, 506, 495 ], "score": 1.0, "content": "Even in the presence of ground-truth proofs for each of the statements in mathlib-train (tactic", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 493, 506, 505 ], "spans": [ { "bbox": [ 105, 493, 506, 505 ], "score": 1.0, "content": "dataset), expert iteration generates data that further improves the performance of the model. The", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 504, 506, 517 ], "spans": [ { "bbox": [ 105, 504, 374, 517 ], "score": 1.0, "content": "number of statements proved in mathlib-train goes from 17390", "type": "text" }, { "bbox": [ 374, 504, 407, 515 ], "score": 0.77, "content": "( 6 7 . 8 \\% )", "type": "inline_equation" }, { "bbox": [ 407, 504, 506, 517 ], "score": 1.0, "content": "at iteration 1 to 19476", "type": "text" } ], "index": 52 }, { "bbox": [ 106, 514, 506, 528 ], "spans": [ { "bbox": [ 106, 515, 140, 526 ], "score": 0.86, "content": "( 7 6 . 0 \\% )", "type": "inline_equation" }, { "bbox": [ 140, 514, 506, 528 ], "score": 1.0, "content": "at iteration 9, while the average proof length of these statements goes from 4.8 to 4.0. We", "type": "text" } ], "index": 53 }, { "bbox": [ 105, 526, 506, 539 ], "spans": [ { "bbox": [ 105, 526, 506, 539 ], "score": 1.0, "content": "hypothesize that this continuously improving performance through expert iteration stems from two", "type": "text" } ], "index": 54 }, { "bbox": [ 105, 536, 506, 551 ], "spans": [ { "bbox": [ 105, 536, 506, 551 ], "score": 1.0, "content": "effects: (i) the model finding new original proofs for the same statements and (ii) the model closing", "type": "text" } ], "index": 55 }, { "bbox": [ 105, 549, 505, 560 ], "spans": [ { "bbox": [ 105, 549, 505, 560 ], "score": 1.0, "content": "marginally harder statements at each iteration – which in turn provides more useful training data for", "type": "text" } ], "index": 56 }, { "bbox": [ 105, 558, 506, 572 ], "spans": [ { "bbox": [ 105, 558, 374, 572 ], "score": 1.0, "content": "the next iteration. By iteration 9, the model is trained on more than", "type": "text" }, { "bbox": [ 374, 559, 394, 570 ], "score": 0.88, "content": "9 0 \\%", "type": "inline_equation" }, { "bbox": [ 395, 558, 506, 572 ], "score": 1.0, "content": "generated data. We present", "type": "text" } ], "index": 57 }, { "bbox": [ 105, 570, 505, 582 ], "spans": [ { "bbox": [ 105, 570, 505, 582 ], "score": 1.0, "content": "in Appendix I a few examples of original proofs found by our models on mathlib-train compared", "type": "text" } ], "index": 58 }, { "bbox": [ 105, 581, 239, 594 ], "spans": [ { "bbox": [ 105, 581, 239, 594 ], "score": 1.0, "content": "with their ground-truth versions.", "type": "text" } ], "index": 59 } ], "index": 54.5, "bbox_fs": [ 105, 481, 506, 594 ] }, { "type": "text", "bbox": [ 107, 597, 505, 653 ], "lines": [ { "bbox": [ 105, 597, 505, 610 ], "spans": [ { "bbox": [ 105, 597, 505, 610 ], "score": 1.0, "content": "To verify our hypothesis that expert iteration is capable of closing a curriculum of increasingly", "type": "text" } ], "index": 60 }, { "bbox": [ 105, 606, 506, 623 ], "spans": [ { "bbox": [ 105, 606, 506, 623 ], "score": 1.0, "content": "difficult problems out of a set of problem statements, and that this capability is independent of having", "type": "text" } ], "index": 61 }, { "bbox": [ 105, 619, 506, 632 ], "spans": [ { "bbox": [ 105, 619, 506, 632 ], "score": 1.0, "content": "access to ground-truth proofs, we propose in the next section to study expert iteration applied to a", "type": "text" } ], "index": 62 }, { "bbox": [ 105, 631, 506, 643 ], "spans": [ { "bbox": [ 105, 631, 506, 643 ], "score": 1.0, "content": "synthetically generated set of problems for which we have fine-grained control on the difficulty of", "type": "text" } ], "index": 63 }, { "bbox": [ 106, 642, 170, 654 ], "spans": [ { "bbox": [ 106, 642, 170, 654 ], "score": 1.0, "content": "each statement.", "type": "text" } ], "index": 64 } ], "index": 62, "bbox_fs": [ 105, 597, 506, 654 ] }, { "type": "title", "bbox": [ 108, 672, 317, 684 ], "lines": [ { "bbox": [ 105, 671, 318, 687 ], "spans": [ { "bbox": [ 105, 671, 318, 687 ], "score": 1.0, "content": "5 STATEMENT CURRICULUM LEARNING", "type": "text" } ], "index": 65 } ], "index": 65 }, { "type": "text", "bbox": [ 108, 699, 505, 732 ], "lines": [ { "bbox": [ 104, 698, 505, 712 ], "spans": [ { "bbox": [ 104, 698, 505, 712 ], "score": 1.0, "content": "In this section we focus on running expert iteration on synthetic statements generated by an inequality", "type": "text" } ], "index": 66 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "generator. The use of synthetic statements enables us to control the difficulty of each statement to", "type": "text" } ], "index": 67 }, { "bbox": [ 105, 720, 506, 733 ], "spans": [ { "bbox": [ 105, 720, 506, 733 ], "score": 1.0, "content": "present evidence that expert iteration can hill-climb the intrinsic difficulty gradient of the resulting set", "type": "text" } ], "index": 68 } ], "index": 67, "bbox_fs": [ 104, 698, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 504, 116 ], "lines": [ { "bbox": [ 105, 81, 506, 95 ], "spans": [ { "bbox": [ 105, 81, 506, 95 ], "score": 1.0, "content": "of statements. In particular, we show that, at fixed compute budget, expert iteration eventually closes", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 505, 106 ], "spans": [ { "bbox": [ 106, 94, 505, 106 ], "score": 1.0, "content": "proofs of hard statements that remain completely out of reach of simply sampling proof searches", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 103, 221, 118 ], "spans": [ { "bbox": [ 106, 103, 221, 118 ], "score": 1.0, "content": "without interleaved training.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "title", "bbox": [ 108, 134, 291, 145 ], "lines": [ { "bbox": [ 105, 133, 293, 146 ], "spans": [ { "bbox": [ 105, 133, 293, 146 ], "score": 1.0, "content": "5.1 SYNTHETIC INEQUALITY GENERATOR", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "text", "bbox": [ 107, 156, 506, 223 ], "lines": [ { "bbox": [ 106, 156, 506, 169 ], "spans": [ { "bbox": [ 106, 156, 506, 169 ], "score": 1.0, "content": "We designed a synthetic inequality statement generator for Lean in the spirit of the INT (Wu et al.,", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 167, 506, 181 ], "spans": [ { "bbox": [ 105, 167, 506, 181 ], "score": 1.0, "content": "2021) generator. The generator consists in generating inequalities from well known inequality theo-", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 177, 507, 191 ], "spans": [ { "bbox": [ 105, 177, 507, 191 ], "score": 1.0, "content": "rems (AM-GM, Trivial inequality, Cauchy-Schwarz, Bernoulli, Young, Hölder) and composing them.", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 189, 506, 202 ], "spans": [ { "bbox": [ 105, 189, 267, 202 ], "score": 1.0, "content": "It is driven by two difficulty parameters:", "type": "text" }, { "bbox": [ 267, 190, 283, 200 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 284, 189, 506, 202 ], "score": 1.0, "content": "which controls depth of composition of inequalities and", "type": "text" } ], "index": 7 }, { "bbox": [ 107, 199, 505, 213 ], "spans": [ { "bbox": [ 107, 200, 122, 212 ], "score": 0.85, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 122, 199, 505, 213 ], "score": 1.0, "content": "which controls the complexity of the input expressions to the composed inequalities. We provide", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 210, 287, 223 ], "spans": [ { "bbox": [ 106, 210, 287, 223 ], "score": 1.0, "content": "details on its implementation in Appendix F.", "type": "text" } ], "index": 9 } ], "index": 6.5 }, { "type": "text", "bbox": [ 107, 228, 505, 294 ], "lines": [ { "bbox": [ 105, 228, 505, 241 ], "spans": [ { "bbox": [ 105, 228, 505, 241 ], "score": 1.0, "content": "Using this generator we generate a curriculum of 5600 inequality statements (for which we don’t have", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 238, 506, 252 ], "spans": [ { "bbox": [ 105, 238, 233, 252 ], "score": 1.0, "content": "proofs), 100 for each values of", "type": "text" }, { "bbox": [ 234, 239, 285, 250 ], "score": 0.92, "content": "0 \\le N _ { S } \\le 7", "type": "inline_equation" }, { "bbox": [ 286, 238, 303, 252 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 304, 239, 357, 250 ], "score": 0.91, "content": "0 \\le N _ { D } \\le 6", "type": "inline_equation" }, { "bbox": [ 357, 238, 506, 252 ], "score": 1.0, "content": ". We denote this set of statements as", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 250, 505, 262 ], "spans": [ { "bbox": [ 105, 250, 505, 262 ], "score": 1.0, "content": "synth-ineq. To bootstrap our models capabilities on this specific task, we also generate 100 statements", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 260, 506, 273 ], "spans": [ { "bbox": [ 105, 260, 176, 273 ], "score": 1.0, "content": "of low difficulty (", "type": "text" }, { "bbox": [ 176, 261, 210, 272 ], "score": 0.91, "content": "N _ { D } = 1", "type": "inline_equation" }, { "bbox": [ 211, 260, 228, 273 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 228, 261, 262, 272 ], "score": 0.91, "content": "N _ { S } = 5", "type": "inline_equation" }, { "bbox": [ 263, 260, 506, 273 ], "score": 1.0, "content": ") and formalize a proof for each of these statements. We refer", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 272, 504, 284 ], "spans": [ { "bbox": [ 105, 272, 504, 284 ], "score": 1.0, "content": "to this dataset as synth-ineq-train. In the rest of this paper we adjunct this training dataset to the", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 283, 271, 295 ], "spans": [ { "bbox": [ 106, 283, 271, 295 ], "score": 1.0, "content": "tactic dataset used to train our models.", "type": "text" } ], "index": 15 } ], "index": 12.5 }, { "type": "title", "bbox": [ 108, 313, 393, 324 ], "lines": [ { "bbox": [ 105, 312, 394, 325 ], "spans": [ { "bbox": [ 105, 312, 394, 325 ], "score": 1.0, "content": "5.2 EXPERT ITERATION ON SYNTHETIC INEQUALITY STATEMENTS", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 334, 505, 368 ], "lines": [ { "bbox": [ 105, 333, 507, 348 ], "spans": [ { "bbox": [ 105, 333, 240, 348 ], "score": 1.0, "content": "In this section we propose to set", "type": "text" }, { "bbox": [ 240, 335, 251, 345 ], "score": 0.82, "content": "S t", "type": "inline_equation" }, { "bbox": [ 251, 333, 507, 348 ], "score": 1.0, "content": "to the union of the statements in mathlib-train and synth-ineq.", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 345, 505, 359 ], "spans": [ { "bbox": [ 105, 345, 208, 359 ], "score": 1.0, "content": "Again, we uniformly set", "type": "text" }, { "bbox": [ 208, 347, 233, 356 ], "score": 0.9, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 234, 345, 268, 359 ], "score": 1.0, "content": "and use", "type": "text" }, { "bbox": [ 268, 346, 279, 357 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 279, 345, 297, 359 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 297, 346, 307, 357 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 308, 345, 505, 359 ], "score": 1.0, "content": "as described in Section 4.3, except that they are", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 357, 257, 369 ], "spans": [ { "bbox": [ 105, 357, 257, 369 ], "score": 1.0, "content": "now also trained on synth-ineq-train.", "type": "text" } ], "index": 19 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 374, 505, 418 ], "lines": [ { "bbox": [ 106, 374, 506, 387 ], "spans": [ { "bbox": [ 106, 374, 506, 387 ], "score": 1.0, "content": "Similarly to the previous section, we report in Figure 3 the cumulative pass rate for two loops, our", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 384, 505, 398 ], "spans": [ { "bbox": [ 105, 384, 505, 398 ], "score": 1.0, "content": "standard expert iteration loop, and a proof search only loop where we do not interleave training", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 396, 506, 409 ], "spans": [ { "bbox": [ 105, 396, 361, 409 ], "score": 1.0, "content": "between iterations. The pass rates are reported split by values of", "type": "text" }, { "bbox": [ 362, 396, 378, 407 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 378, 396, 450, 409 ], "score": 1.0, "content": "(pooling together", "type": "text" }, { "bbox": [ 450, 396, 502, 407 ], "score": 0.9, "content": "0 \\le N _ { S } \\le 7", "type": "inline_equation" }, { "bbox": [ 503, 396, 506, 409 ], "score": 1.0, "content": ")", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 405, 313, 420 ], "spans": [ { "bbox": [ 105, 405, 313, 420 ], "score": 1.0, "content": "which we found to be the main driver for difficulty.", "type": "text" } ], "index": 23 } ], "index": 21.5 }, { "type": "image", "bbox": [ 133, 436, 477, 538 ], "blocks": [ { "type": "image_body", "bbox": [ 133, 436, 477, 538 ], "group_id": 0, "lines": [ { "bbox": [ 133, 436, 477, 538 ], "spans": [ { "bbox": [ 133, 436, 477, 538 ], "score": 0.966, "type": "image", "image_path": "33f55772b311eab974522ad234fc4f4e9c0106ba9fd7089993d50b26eaa7d7d2.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 133, 436, 477, 470.0 ], "spans": [], "index": 24 }, { "bbox": [ 133, 470.0, 477, 504.0 ], "spans": [], "index": 25 }, { "bbox": [ 133, 504.0, 477, 538.0 ], "spans": [], "index": 26 } ] }, { "type": "image_caption", "bbox": [ 106, 553, 505, 587 ], "group_id": 0, "lines": [ { "bbox": [ 105, 552, 505, 567 ], "spans": [ { "bbox": [ 105, 552, 505, 567 ], "score": 1.0, "content": "Figure 3: Cumulative pass rate for our expert iteration loop as well as a sample only loop where we", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 563, 505, 578 ], "spans": [ { "bbox": [ 105, 563, 451, 578 ], "score": 1.0, "content": "skip re-training the model between iterations. Pass rates are reported for each value of", "type": "text" }, { "bbox": [ 451, 565, 468, 576 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 468, 563, 505, 578 ], "score": 1.0, "content": "(pooling", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 575, 200, 588 ], "spans": [ { "bbox": [ 106, 575, 141, 588 ], "score": 1.0, "content": "together", "type": "text" }, { "bbox": [ 142, 575, 194, 587 ], "score": 0.91, "content": "0 \\le N _ { S } \\le 7", "type": "inline_equation" }, { "bbox": [ 194, 575, 200, 588 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 29 } ], "index": 28 } ], "index": 26.5 }, { "type": "text", "bbox": [ 107, 615, 505, 682 ], "lines": [ { "bbox": [ 105, 616, 506, 629 ], "spans": [ { "bbox": [ 105, 616, 506, 629 ], "score": 1.0, "content": "Despite the challenging nature of these synthetic inequalities, Figure 3 demonstrates that expert", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 627, 506, 640 ], "spans": [ { "bbox": [ 105, 627, 506, 640 ], "score": 1.0, "content": "iteration is capable of learning the intrinsic curriculum induced by synth-ineq. In particular, expert", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 638, 505, 651 ], "spans": [ { "bbox": [ 105, 638, 325, 651 ], "score": 1.0, "content": "iteration is capable of closing 6 problems of difficulty", "type": "text" }, { "bbox": [ 326, 638, 361, 649 ], "score": 0.91, "content": "N _ { D } = 6", "type": "inline_equation" }, { "bbox": [ 361, 638, 505, 651 ], "score": 1.0, "content": "without having been provided with", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 648, 506, 663 ], "spans": [ { "bbox": [ 105, 648, 389, 663 ], "score": 1.0, "content": "any seed ground-truth proof for this difficulty level. Note that difficulty", "type": "text" }, { "bbox": [ 390, 649, 424, 660 ], "score": 0.91, "content": "N _ { D } = 6", "type": "inline_equation" }, { "bbox": [ 425, 648, 506, 663 ], "score": 1.0, "content": "remains completely", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 659, 506, 674 ], "spans": [ { "bbox": [ 105, 659, 506, 674 ], "score": 1.0, "content": "out of reach of simply scaling the number of attempts per statements (the sample only loop remaining", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 671, 203, 683 ], "spans": [ { "bbox": [ 106, 671, 161, 683 ], "score": 1.0, "content": "stuck at 0 for", "type": "text" }, { "bbox": [ 162, 671, 196, 682 ], "score": 0.89, "content": "N _ { D } = 6", "type": "inline_equation" }, { "bbox": [ 197, 671, 203, 683 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 35 } ], "index": 32.5 }, { "type": "text", "bbox": [ 107, 687, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "This confirms on our synthetic statements dataset synth-ineq that not only expert iteration is capable", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 505, 711 ], "score": 1.0, "content": "of learning the curricula occurring in a set of statements, but this process also enables the emergence", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 709, 507, 723 ], "spans": [ { "bbox": [ 105, 709, 507, 723 ], "score": 1.0, "content": "of new capabilities without the need for ground-truth proofs (ability to close, highly challenging,", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 721, 233, 734 ], "spans": [ { "bbox": [ 105, 721, 233, 734 ], "score": 1.0, "content": "deeply composed inequalities).", "type": "text" } ], "index": 39 } ], "index": 37.5 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 25, 293, 38 ], "spans": [ { "bbox": [ 106, 25, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "7", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 504, 116 ], "lines": [ { "bbox": [ 105, 81, 506, 95 ], "spans": [ { "bbox": [ 105, 81, 506, 95 ], "score": 1.0, "content": "of statements. In particular, we show that, at fixed compute budget, expert iteration eventually closes", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 94, 505, 106 ], "spans": [ { "bbox": [ 106, 94, 505, 106 ], "score": 1.0, "content": "proofs of hard statements that remain completely out of reach of simply sampling proof searches", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 103, 221, 118 ], "spans": [ { "bbox": [ 106, 103, 221, 118 ], "score": 1.0, "content": "without interleaved training.", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 105, 81, 506, 118 ] }, { "type": "title", "bbox": [ 108, 134, 291, 145 ], "lines": [ { "bbox": [ 105, 133, 293, 146 ], "spans": [ { "bbox": [ 105, 133, 293, 146 ], "score": 1.0, "content": "5.1 SYNTHETIC INEQUALITY GENERATOR", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "text", "bbox": [ 107, 156, 506, 223 ], "lines": [ { "bbox": [ 106, 156, 506, 169 ], "spans": [ { "bbox": [ 106, 156, 506, 169 ], "score": 1.0, "content": "We designed a synthetic inequality statement generator for Lean in the spirit of the INT (Wu et al.,", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 167, 506, 181 ], "spans": [ { "bbox": [ 105, 167, 506, 181 ], "score": 1.0, "content": "2021) generator. The generator consists in generating inequalities from well known inequality theo-", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 177, 507, 191 ], "spans": [ { "bbox": [ 105, 177, 507, 191 ], "score": 1.0, "content": "rems (AM-GM, Trivial inequality, Cauchy-Schwarz, Bernoulli, Young, Hölder) and composing them.", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 189, 506, 202 ], "spans": [ { "bbox": [ 105, 189, 267, 202 ], "score": 1.0, "content": "It is driven by two difficulty parameters:", "type": "text" }, { "bbox": [ 267, 190, 283, 200 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 284, 189, 506, 202 ], "score": 1.0, "content": "which controls depth of composition of inequalities and", "type": "text" } ], "index": 7 }, { "bbox": [ 107, 199, 505, 213 ], "spans": [ { "bbox": [ 107, 200, 122, 212 ], "score": 0.85, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 122, 199, 505, 213 ], "score": 1.0, "content": "which controls the complexity of the input expressions to the composed inequalities. We provide", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 210, 287, 223 ], "spans": [ { "bbox": [ 106, 210, 287, 223 ], "score": 1.0, "content": "details on its implementation in Appendix F.", "type": "text" } ], "index": 9 } ], "index": 6.5, "bbox_fs": [ 105, 156, 507, 223 ] }, { "type": "text", "bbox": [ 107, 228, 505, 294 ], "lines": [ { "bbox": [ 105, 228, 505, 241 ], "spans": [ { "bbox": [ 105, 228, 505, 241 ], "score": 1.0, "content": "Using this generator we generate a curriculum of 5600 inequality statements (for which we don’t have", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 238, 506, 252 ], "spans": [ { "bbox": [ 105, 238, 233, 252 ], "score": 1.0, "content": "proofs), 100 for each values of", "type": "text" }, { "bbox": [ 234, 239, 285, 250 ], "score": 0.92, "content": "0 \\le N _ { S } \\le 7", "type": "inline_equation" }, { "bbox": [ 286, 238, 303, 252 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 304, 239, 357, 250 ], "score": 0.91, "content": "0 \\le N _ { D } \\le 6", "type": "inline_equation" }, { "bbox": [ 357, 238, 506, 252 ], "score": 1.0, "content": ". We denote this set of statements as", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 250, 505, 262 ], "spans": [ { "bbox": [ 105, 250, 505, 262 ], "score": 1.0, "content": "synth-ineq. To bootstrap our models capabilities on this specific task, we also generate 100 statements", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 260, 506, 273 ], "spans": [ { "bbox": [ 105, 260, 176, 273 ], "score": 1.0, "content": "of low difficulty (", "type": "text" }, { "bbox": [ 176, 261, 210, 272 ], "score": 0.91, "content": "N _ { D } = 1", "type": "inline_equation" }, { "bbox": [ 211, 260, 228, 273 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 228, 261, 262, 272 ], "score": 0.91, "content": "N _ { S } = 5", "type": "inline_equation" }, { "bbox": [ 263, 260, 506, 273 ], "score": 1.0, "content": ") and formalize a proof for each of these statements. We refer", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 272, 504, 284 ], "spans": [ { "bbox": [ 105, 272, 504, 284 ], "score": 1.0, "content": "to this dataset as synth-ineq-train. In the rest of this paper we adjunct this training dataset to the", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 283, 271, 295 ], "spans": [ { "bbox": [ 106, 283, 271, 295 ], "score": 1.0, "content": "tactic dataset used to train our models.", "type": "text" } ], "index": 15 } ], "index": 12.5, "bbox_fs": [ 105, 228, 506, 295 ] }, { "type": "title", "bbox": [ 108, 313, 393, 324 ], "lines": [ { "bbox": [ 105, 312, 394, 325 ], "spans": [ { "bbox": [ 105, 312, 394, 325 ], "score": 1.0, "content": "5.2 EXPERT ITERATION ON SYNTHETIC INEQUALITY STATEMENTS", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 334, 505, 368 ], "lines": [ { "bbox": [ 105, 333, 507, 348 ], "spans": [ { "bbox": [ 105, 333, 240, 348 ], "score": 1.0, "content": "In this section we propose to set", "type": "text" }, { "bbox": [ 240, 335, 251, 345 ], "score": 0.82, "content": "S t", "type": "inline_equation" }, { "bbox": [ 251, 333, 507, 348 ], "score": 1.0, "content": "to the union of the statements in mathlib-train and synth-ineq.", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 345, 505, 359 ], "spans": [ { "bbox": [ 105, 345, 208, 359 ], "score": 1.0, "content": "Again, we uniformly set", "type": "text" }, { "bbox": [ 208, 347, 233, 356 ], "score": 0.9, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 234, 345, 268, 359 ], "score": 1.0, "content": "and use", "type": "text" }, { "bbox": [ 268, 346, 279, 357 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 279, 345, 297, 359 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 297, 346, 307, 357 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 308, 345, 505, 359 ], "score": 1.0, "content": "as described in Section 4.3, except that they are", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 357, 257, 369 ], "spans": [ { "bbox": [ 105, 357, 257, 369 ], "score": 1.0, "content": "now also trained on synth-ineq-train.", "type": "text" } ], "index": 19 } ], "index": 18, "bbox_fs": [ 105, 333, 507, 369 ] }, { "type": "text", "bbox": [ 107, 374, 505, 418 ], "lines": [ { "bbox": [ 106, 374, 506, 387 ], "spans": [ { "bbox": [ 106, 374, 506, 387 ], "score": 1.0, "content": "Similarly to the previous section, we report in Figure 3 the cumulative pass rate for two loops, our", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 384, 505, 398 ], "spans": [ { "bbox": [ 105, 384, 505, 398 ], "score": 1.0, "content": "standard expert iteration loop, and a proof search only loop where we do not interleave training", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 396, 506, 409 ], "spans": [ { "bbox": [ 105, 396, 361, 409 ], "score": 1.0, "content": "between iterations. The pass rates are reported split by values of", "type": "text" }, { "bbox": [ 362, 396, 378, 407 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 378, 396, 450, 409 ], "score": 1.0, "content": "(pooling together", "type": "text" }, { "bbox": [ 450, 396, 502, 407 ], "score": 0.9, "content": "0 \\le N _ { S } \\le 7", "type": "inline_equation" }, { "bbox": [ 503, 396, 506, 409 ], "score": 1.0, "content": ")", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 405, 313, 420 ], "spans": [ { "bbox": [ 105, 405, 313, 420 ], "score": 1.0, "content": "which we found to be the main driver for difficulty.", "type": "text" } ], "index": 23 } ], "index": 21.5, "bbox_fs": [ 105, 374, 506, 420 ] }, { "type": "image", "bbox": [ 133, 436, 477, 538 ], "blocks": [ { "type": "image_body", "bbox": [ 133, 436, 477, 538 ], "group_id": 0, "lines": [ { "bbox": [ 133, 436, 477, 538 ], "spans": [ { "bbox": [ 133, 436, 477, 538 ], "score": 0.966, "type": "image", "image_path": "33f55772b311eab974522ad234fc4f4e9c0106ba9fd7089993d50b26eaa7d7d2.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 133, 436, 477, 470.0 ], "spans": [], "index": 24 }, { "bbox": [ 133, 470.0, 477, 504.0 ], "spans": [], "index": 25 }, { "bbox": [ 133, 504.0, 477, 538.0 ], "spans": [], "index": 26 } ] }, { "type": "image_caption", "bbox": [ 106, 553, 505, 587 ], "group_id": 0, "lines": [ { "bbox": [ 105, 552, 505, 567 ], "spans": [ { "bbox": [ 105, 552, 505, 567 ], "score": 1.0, "content": "Figure 3: Cumulative pass rate for our expert iteration loop as well as a sample only loop where we", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 563, 505, 578 ], "spans": [ { "bbox": [ 105, 563, 451, 578 ], "score": 1.0, "content": "skip re-training the model between iterations. Pass rates are reported for each value of", "type": "text" }, { "bbox": [ 451, 565, 468, 576 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 468, 563, 505, 578 ], "score": 1.0, "content": "(pooling", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 575, 200, 588 ], "spans": [ { "bbox": [ 106, 575, 141, 588 ], "score": 1.0, "content": "together", "type": "text" }, { "bbox": [ 142, 575, 194, 587 ], "score": 0.91, "content": "0 \\le N _ { S } \\le 7", "type": "inline_equation" }, { "bbox": [ 194, 575, 200, 588 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 29 } ], "index": 28 } ], "index": 26.5 }, { "type": "text", "bbox": [ 107, 615, 505, 682 ], "lines": [ { "bbox": [ 105, 616, 506, 629 ], "spans": [ { "bbox": [ 105, 616, 506, 629 ], "score": 1.0, "content": "Despite the challenging nature of these synthetic inequalities, Figure 3 demonstrates that expert", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 627, 506, 640 ], "spans": [ { "bbox": [ 105, 627, 506, 640 ], "score": 1.0, "content": "iteration is capable of learning the intrinsic curriculum induced by synth-ineq. In particular, expert", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 638, 505, 651 ], "spans": [ { "bbox": [ 105, 638, 325, 651 ], "score": 1.0, "content": "iteration is capable of closing 6 problems of difficulty", "type": "text" }, { "bbox": [ 326, 638, 361, 649 ], "score": 0.91, "content": "N _ { D } = 6", "type": "inline_equation" }, { "bbox": [ 361, 638, 505, 651 ], "score": 1.0, "content": "without having been provided with", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 648, 506, 663 ], "spans": [ { "bbox": [ 105, 648, 389, 663 ], "score": 1.0, "content": "any seed ground-truth proof for this difficulty level. Note that difficulty", "type": "text" }, { "bbox": [ 390, 649, 424, 660 ], "score": 0.91, "content": "N _ { D } = 6", "type": "inline_equation" }, { "bbox": [ 425, 648, 506, 663 ], "score": 1.0, "content": "remains completely", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 659, 506, 674 ], "spans": [ { "bbox": [ 105, 659, 506, 674 ], "score": 1.0, "content": "out of reach of simply scaling the number of attempts per statements (the sample only loop remaining", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 671, 203, 683 ], "spans": [ { "bbox": [ 106, 671, 161, 683 ], "score": 1.0, "content": "stuck at 0 for", "type": "text" }, { "bbox": [ 162, 671, 196, 682 ], "score": 0.89, "content": "N _ { D } = 6", "type": "inline_equation" }, { "bbox": [ 197, 671, 203, 683 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 35 } ], "index": 32.5, "bbox_fs": [ 105, 616, 506, 683 ] }, { "type": "text", "bbox": [ 107, 687, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "This confirms on our synthetic statements dataset synth-ineq that not only expert iteration is capable", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 505, 711 ], "score": 1.0, "content": "of learning the curricula occurring in a set of statements, but this process also enables the emergence", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 709, 507, 723 ], "spans": [ { "bbox": [ 105, 709, 507, 723 ], "score": 1.0, "content": "of new capabilities without the need for ground-truth proofs (ability to close, highly challenging,", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 721, 233, 734 ], "spans": [ { "bbox": [ 105, 721, 233, 734 ], "score": 1.0, "content": "deeply composed inequalities).", "type": "text" } ], "index": 39 } ], "index": 37.5, "bbox_fs": [ 105, 687, 507, 734 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 81, 232, 94 ], "lines": [ { "bbox": [ 105, 79, 234, 95 ], "spans": [ { "bbox": [ 105, 79, 234, 95 ], "score": 1.0, "content": "6 TARGETING miniF2F", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 105, 505, 138 ], "lines": [ { "bbox": [ 106, 106, 504, 117 ], "spans": [ { "bbox": [ 106, 106, 504, 117 ], "score": 1.0, "content": "Motivated by the results from Section 5, we curated and manually formalized a set of math exercises", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 117, 505, 128 ], "spans": [ { "bbox": [ 106, 117, 177, 128 ], "score": 1.0, "content": "denoted as miniF", "type": "text" }, { "bbox": [ 177, 117, 190, 127 ], "score": 0.64, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 190, 117, 342, 128 ], "score": 1.0, "content": "-curriculum to target miniF2F. miniF", "type": "text" }, { "bbox": [ 342, 117, 355, 127 ], "score": 0.49, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 355, 117, 505, 128 ], "score": 1.0, "content": "-curriculum contains 327 statements", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 128, 433, 140 ], "spans": [ { "bbox": [ 106, 128, 433, 140 ], "score": 1.0, "content": "from various sources, with their provenance and analysis detailed in Appendix G.", "type": "text" } ], "index": 3 } ], "index": 2 }, { "type": "text", "bbox": [ 107, 144, 505, 188 ], "lines": [ { "bbox": [ 105, 144, 505, 157 ], "spans": [ { "bbox": [ 105, 144, 130, 157 ], "score": 1.0, "content": "miniF", "type": "text" }, { "bbox": [ 130, 145, 143, 155 ], "score": 0.47, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 144, 144, 505, 157 ], "score": 1.0, "content": "statements being quite out of distribution compared to mathlib statements (which typically", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 155, 506, 168 ], "spans": [ { "bbox": [ 105, 155, 506, 168 ], "score": 1.0, "content": "are generic theorems and lemmas), we hypothesized that if the difficulty of miniF2F-curriculum was", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 167, 506, 178 ], "spans": [ { "bbox": [ 106, 167, 506, 178 ], "score": 1.0, "content": "made varied enough, expert iteration could potentially leverage it to effectively shift our models’", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 177, 454, 189 ], "spans": [ { "bbox": [ 106, 177, 215, 189 ], "score": 1.0, "content": "distribution closer to miniF", "type": "text" }, { "bbox": [ 215, 178, 228, 187 ], "score": 0.56, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 228, 177, 454, 189 ], "score": 1.0, "content": "’s, and in turn, improve their eventual performance on it.", "type": "text" } ], "index": 7 } ], "index": 5.5 }, { "type": "title", "bbox": [ 107, 201, 229, 212 ], "lines": [ { "bbox": [ 106, 201, 230, 214 ], "spans": [ { "bbox": [ 106, 201, 230, 214 ], "score": 1.0, "content": "6.1 TRANSFER TO miniF2F", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 108, 222, 504, 255 ], "lines": [ { "bbox": [ 104, 221, 506, 236 ], "spans": [ { "bbox": [ 104, 221, 246, 236 ], "score": 1.0, "content": "In this section we propose to set", "type": "text" }, { "bbox": [ 247, 223, 258, 232 ], "score": 0.81, "content": "S t", "type": "inline_equation" }, { "bbox": [ 258, 221, 506, 236 ], "score": 1.0, "content": "to the union of the statements in mathlib-train, synth-ineq", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 233, 505, 245 ], "spans": [ { "bbox": [ 106, 233, 288, 245 ], "score": 1.0, "content": "and miniF2F-curriculum. We uniformly set", "type": "text" }, { "bbox": [ 288, 234, 314, 244 ], "score": 0.89, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 314, 233, 465, 245 ], "score": 1.0, "content": "on mathlib-train and synth-ineq and", "type": "text" }, { "bbox": [ 465, 234, 491, 244 ], "score": 0.89, "content": "a = 8", "type": "inline_equation" }, { "bbox": [ 492, 233, 505, 245 ], "score": 1.0, "content": "on", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 245, 368, 255 ], "spans": [ { "bbox": [ 106, 245, 130, 255 ], "score": 1.0, "content": "miniF", "type": "text" }, { "bbox": [ 130, 245, 143, 254 ], "score": 0.61, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 143, 245, 223, 255 ], "score": 1.0, "content": "-curriculum and use", "type": "text" }, { "bbox": [ 224, 245, 234, 255 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 235, 245, 252, 255 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 253, 245, 263, 255 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 263, 245, 368, 255 ], "score": 1.0, "content": "as described in Section 5.", "type": "text" } ], "index": 11 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 261, 505, 327 ], "lines": [ { "bbox": [ 106, 261, 505, 273 ], "spans": [ { "bbox": [ 106, 261, 469, 273 ], "score": 1.0, "content": "Similarly to previous sections, we report in Figure 4 (left) the cumulative pass rate on miniF", "type": "text" }, { "bbox": [ 469, 262, 481, 271 ], "score": 0.43, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 482, 261, 505, 273 ], "score": 1.0, "content": "-valid", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 273, 505, 284 ], "spans": [ { "bbox": [ 106, 273, 505, 284 ], "score": 1.0, "content": "of our full curriculum expert iteration loop and compare them with the mathlib-train only expert", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 283, 506, 296 ], "spans": [ { "bbox": [ 105, 283, 506, 296 ], "score": 1.0, "content": "iteration from Section 4.5. Since more compute is deployed in our full-curriculum loop (more", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 294, 506, 307 ], "spans": [ { "bbox": [ 105, 294, 343, 307 ], "score": 1.0, "content": "statements), we also report a mathlib-train only loop taking", "type": "text" }, { "bbox": [ 344, 294, 369, 304 ], "score": 0.9, "content": "a = 2", "type": "inline_equation" }, { "bbox": [ 369, 294, 506, 307 ], "score": 1.0, "content": ". At the end of the expert iteration,", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 304, 506, 318 ], "spans": [ { "bbox": [ 105, 304, 273, 318 ], "score": 1.0, "content": "100 out of the 327 statements from miniF", "type": "text" }, { "bbox": [ 274, 306, 287, 315 ], "score": 0.32, "content": "2 F .", "type": "inline_equation" }, { "bbox": [ 287, 304, 506, 318 ], "score": 1.0, "content": "-curriculum end up being closed, suggesting a lack of", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 316, 316, 328 ], "spans": [ { "bbox": [ 106, 316, 316, 328 ], "score": 1.0, "content": "density in our manually formalized set of statement.", "type": "text" } ], "index": 17 } ], "index": 14.5 }, { "type": "text", "bbox": [ 106, 333, 506, 402 ], "lines": [ { "bbox": [ 105, 332, 506, 346 ], "spans": [ { "bbox": [ 105, 332, 269, 346 ], "score": 1.0, "content": "We also report in Figure 4 (right) the pass", "type": "text" }, { "bbox": [ 269, 333, 283, 343 ], "score": 0.69, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 283, 332, 318, 346 ], "score": 1.0, "content": "and pass", "type": "text" }, { "bbox": [ 319, 333, 333, 343 ], "score": 0.63, "content": "@ 8", "type": "inline_equation" }, { "bbox": [ 334, 332, 506, 346 ], "score": 1.0, "content": "for our full curriculum expert iteration loop.", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 344, 505, 356 ], "spans": [ { "bbox": [ 106, 344, 505, 356 ], "score": 1.0, "content": "The steady improvement on miniF2F-valid shows that the expert iteration procedure we propose does", "type": "text" } ], "index": 19 }, { "bbox": [ 104, 353, 506, 369 ], "spans": [ { "bbox": [ 104, 353, 506, 369 ], "score": 1.0, "content": "not overfit on the statements that compose the curriculum it uses. Despite the potential inefficiency", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 366, 506, 378 ], "spans": [ { "bbox": [ 106, 366, 506, 378 ], "score": 1.0, "content": "of our curriculum, the improved performance associated with its use demonstrates, as hypothesized,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 376, 506, 390 ], "spans": [ { "bbox": [ 105, 376, 251, 390 ], "score": 1.0, "content": "an effective transfer between miniF", "type": "text" }, { "bbox": [ 251, 377, 265, 387 ], "score": 0.43, "content": "{ } ^ { 7 2 F }", "type": "inline_equation" }, { "bbox": [ 265, 376, 506, 390 ], "score": 1.0, "content": "-curriculum, synth-ineq and miniF2F-valid through expert", "type": "text" } ], "index": 22 }, { "bbox": [ 103, 387, 404, 404 ], "spans": [ { "bbox": [ 103, 387, 384, 404 ], "score": 1.0, "content": "iteration. We will denote the fully iterated model from this section as", "type": "text" }, { "bbox": [ 384, 387, 400, 402 ], "score": 0.91, "content": "\\theta _ { 9 } ^ { f u l l }", "type": "inline_equation" }, { "bbox": [ 400, 387, 404, 404 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 23 } ], "index": 20.5 }, { "type": "image", "bbox": [ 123, 414, 489, 516 ], "blocks": [ { "type": "image_body", "bbox": [ 123, 414, 489, 516 ], "group_id": 0, "lines": [ { "bbox": [ 123, 414, 489, 516 ], "spans": [ { "bbox": [ 123, 414, 489, 516 ], "score": 0.958, "type": "image", "image_path": "04a8b08c5b09b13442660861e017e088784c96f4097bff3f86db3eb43c8ba58a.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 123, 414, 489, 448.0 ], "spans": [], "index": 24 }, { "bbox": [ 123, 448.0, 489, 482.0 ], "spans": [], "index": 25 }, { "bbox": [ 123, 482.0, 489, 516.0 ], "spans": [], "index": 26 } ] }, { "type": "image_caption", "bbox": [ 106, 531, 506, 619 ], "group_id": 0, "lines": [ { "bbox": [ 106, 531, 506, 544 ], "spans": [ { "bbox": [ 106, 531, 294, 544 ], "score": 1.0, "content": "Figure 4: Left: cumulative pass rate on miniF", "type": "text" }, { "bbox": [ 294, 532, 307, 541 ], "score": 0.26, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 307, 531, 506, 544 ], "score": 1.0, "content": "-valid for our expert iteration loop using our full", "type": "text" } ], "index": 27 }, { "bbox": [ 104, 541, 506, 555 ], "spans": [ { "bbox": [ 104, 541, 294, 555 ], "score": 1.0, "content": "curriculum (mathlib-train, synth-ineq and miniF", "type": "text" }, { "bbox": [ 295, 542, 307, 552 ], "score": 0.47, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 308, 541, 506, 555 ], "score": 1.0, "content": "-curriculum) compared to the expert iteration loop", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 552, 506, 565 ], "spans": [ { "bbox": [ 105, 552, 410, 565 ], "score": 1.0, "content": "from Section 4.5. The total number of attempts per iteration in our full loop is", "type": "text" }, { "bbox": [ 410, 553, 506, 564 ], "score": 0.9, "content": "2 5 k + 5 . 6 k + 8 * 3 2 7 \\approx", "type": "inline_equation" } ], "index": 29 }, { "bbox": [ 106, 564, 506, 576 ], "spans": [ { "bbox": [ 106, 564, 131, 574 ], "score": 0.85, "content": "3 3 . 2 k", "type": "inline_equation" }, { "bbox": [ 131, 564, 480, 576 ], "score": 1.0, "content": ", which means the total compute deployed is higher than in the mathlib-train only loop", "type": "text" }, { "bbox": [ 480, 564, 502, 575 ], "score": 0.81, "content": "( 2 5 k )", "type": "inline_equation" }, { "bbox": [ 503, 564, 506, 576 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 575, 506, 588 ], "spans": [ { "bbox": [ 105, 575, 381, 588 ], "score": 1.0, "content": "We therefore also report in dotted a mathlib-train only loop, taking", "type": "text" }, { "bbox": [ 382, 576, 407, 585 ], "score": 0.88, "content": "a = 2", "type": "inline_equation" }, { "bbox": [ 407, 575, 506, 588 ], "score": 1.0, "content": ", whose total number of", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 586, 505, 598 ], "spans": [ { "bbox": [ 105, 586, 202, 598 ], "score": 1.0, "content": "attempts per iteration is", "type": "text" }, { "bbox": [ 202, 586, 230, 596 ], "score": 0.88, "content": "\\approx 5 0 k", "type": "inline_equation" }, { "bbox": [ 230, 586, 283, 598 ], "score": 1.0, "content": ". Right: pass", "type": "text" }, { "bbox": [ 283, 586, 297, 596 ], "score": 0.45, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 298, 586, 363, 598 ], "score": 1.0, "content": "(plain) and pass", "type": "text" }, { "bbox": [ 363, 586, 378, 596 ], "score": 0.61, "content": "@ 8", "type": "inline_equation" }, { "bbox": [ 378, 586, 505, 598 ], "score": 1.0, "content": "(dotted) for our expert iteration", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 597, 506, 610 ], "spans": [ { "bbox": [ 104, 597, 374, 610 ], "score": 1.0, "content": "loop using our full curriculum (mathlib-train, synth-ineq and miniF", "type": "text" }, { "bbox": [ 375, 597, 387, 607 ], "score": 0.36, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 387, 597, 506, 610 ], "score": 1.0, "content": "-curriculum) compared to the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 608, 262, 620 ], "spans": [ { "bbox": [ 105, 608, 262, 620 ], "score": 1.0, "content": "expert iteration loop from Section 4.5.", "type": "text" } ], "index": 34 } ], "index": 30.5 } ], "index": 27.75 }, { "type": "title", "bbox": [ 107, 637, 170, 649 ], "lines": [ { "bbox": [ 106, 636, 171, 650 ], "spans": [ { "bbox": [ 106, 636, 171, 650 ], "score": 1.0, "content": "6.2 RESULTS", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 107, 658, 505, 704 ], "lines": [ { "bbox": [ 105, 658, 505, 671 ], "spans": [ { "bbox": [ 105, 658, 505, 671 ], "score": 1.0, "content": "We report in Table 2 the pass rates on mathlib-{valid, test} and miniF2F-{valid, test} for the models", "type": "text" } ], "index": 36 }, { "bbox": [ 103, 666, 505, 687 ], "spans": [ { "bbox": [ 103, 666, 248, 687 ], "score": 1.0, "content": "trained in previous sections, namely", "type": "text" }, { "bbox": [ 248, 671, 259, 682 ], "score": 0.82, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 259, 666, 310, 687 ], "score": 1.0, "content": ", θmathlib9 , and", "type": "text" }, { "bbox": [ 310, 669, 326, 683 ], "score": 0.9, "content": "\\theta _ { 9 } ^ { f u l l }", "type": "inline_equation" }, { "bbox": [ 326, 666, 383, 687 ], "score": 1.0, "content": ". We achieve a", "type": "text" }, { "bbox": [ 384, 670, 411, 682 ], "score": 0.87, "content": "4 7 . 3 \\%", "type": "inline_equation" }, { "bbox": [ 411, 666, 474, 687 ], "score": 1.0, "content": "pass rate (using", "type": "text" }, { "bbox": [ 474, 671, 505, 681 ], "score": 0.87, "content": "a = 6 4", "type": "inline_equation" } ], "index": 37 }, { "bbox": [ 105, 681, 505, 695 ], "spans": [ { "bbox": [ 105, 681, 181, 695 ], "score": 1.0, "content": "attempts) on miniF", "type": "text" }, { "bbox": [ 181, 682, 194, 692 ], "score": 0.62, "content": "2 F .", "type": "inline_equation" }, { "bbox": [ 194, 681, 241, 695 ], "score": 1.0, "content": "-valid and a", "type": "text" }, { "bbox": [ 241, 682, 268, 692 ], "score": 0.87, "content": "3 6 . 6 \\%", "type": "inline_equation" }, { "bbox": [ 268, 681, 505, 695 ], "score": 1.0, "content": "pass rate on miniF2F-test, substantially improving from the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 693, 288, 705 ], "spans": [ { "bbox": [ 105, 693, 288, 705 ], "score": 1.0, "content": "previous state-of-the-art (Zheng et al., 2022).", "type": "text" } ], "index": 39 } ], "index": 37.5 }, { "type": "text", "bbox": [ 107, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 710, 505, 721 ], "spans": [ { "bbox": [ 106, 710, 505, 721 ], "score": 1.0, "content": "These results include the resolution of 26 AMC12 problems, 6 AIME problems and 2 IMO-adapted", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "problems. Out of these statements, 4 AMC12 problems (amc12b_2020_p5, amc12a_2009_p9,", "type": "text" } ], "index": 41 } ], "index": 40.5 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 750, 309, 761 ], "spans": [ { "bbox": [ 302, 750, 309, 761 ], "score": 1.0, "content": "8", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 108, 81, 232, 94 ], "lines": [ { "bbox": [ 105, 79, 234, 95 ], "spans": [ { "bbox": [ 105, 79, 234, 95 ], "score": 1.0, "content": "6 TARGETING miniF2F", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 105, 505, 138 ], "lines": [ { "bbox": [ 106, 106, 504, 117 ], "spans": [ { "bbox": [ 106, 106, 504, 117 ], "score": 1.0, "content": "Motivated by the results from Section 5, we curated and manually formalized a set of math exercises", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 117, 505, 128 ], "spans": [ { "bbox": [ 106, 117, 177, 128 ], "score": 1.0, "content": "denoted as miniF", "type": "text" }, { "bbox": [ 177, 117, 190, 127 ], "score": 0.64, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 190, 117, 342, 128 ], "score": 1.0, "content": "-curriculum to target miniF2F. miniF", "type": "text" }, { "bbox": [ 342, 117, 355, 127 ], "score": 0.49, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 355, 117, 505, 128 ], "score": 1.0, "content": "-curriculum contains 327 statements", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 128, 433, 140 ], "spans": [ { "bbox": [ 106, 128, 433, 140 ], "score": 1.0, "content": "from various sources, with their provenance and analysis detailed in Appendix G.", "type": "text" } ], "index": 3 } ], "index": 2, "bbox_fs": [ 106, 106, 505, 140 ] }, { "type": "text", "bbox": [ 107, 144, 505, 188 ], "lines": [ { "bbox": [ 105, 144, 505, 157 ], "spans": [ { "bbox": [ 105, 144, 130, 157 ], "score": 1.0, "content": "miniF", "type": "text" }, { "bbox": [ 130, 145, 143, 155 ], "score": 0.47, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 144, 144, 505, 157 ], "score": 1.0, "content": "statements being quite out of distribution compared to mathlib statements (which typically", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 155, 506, 168 ], "spans": [ { "bbox": [ 105, 155, 506, 168 ], "score": 1.0, "content": "are generic theorems and lemmas), we hypothesized that if the difficulty of miniF2F-curriculum was", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 167, 506, 178 ], "spans": [ { "bbox": [ 106, 167, 506, 178 ], "score": 1.0, "content": "made varied enough, expert iteration could potentially leverage it to effectively shift our models’", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 177, 454, 189 ], "spans": [ { "bbox": [ 106, 177, 215, 189 ], "score": 1.0, "content": "distribution closer to miniF", "type": "text" }, { "bbox": [ 215, 178, 228, 187 ], "score": 0.56, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 228, 177, 454, 189 ], "score": 1.0, "content": "’s, and in turn, improve their eventual performance on it.", "type": "text" } ], "index": 7 } ], "index": 5.5, "bbox_fs": [ 105, 144, 506, 189 ] }, { "type": "title", "bbox": [ 107, 201, 229, 212 ], "lines": [ { "bbox": [ 106, 201, 230, 214 ], "spans": [ { "bbox": [ 106, 201, 230, 214 ], "score": 1.0, "content": "6.1 TRANSFER TO miniF2F", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 108, 222, 504, 255 ], "lines": [ { "bbox": [ 104, 221, 506, 236 ], "spans": [ { "bbox": [ 104, 221, 246, 236 ], "score": 1.0, "content": "In this section we propose to set", "type": "text" }, { "bbox": [ 247, 223, 258, 232 ], "score": 0.81, "content": "S t", "type": "inline_equation" }, { "bbox": [ 258, 221, 506, 236 ], "score": 1.0, "content": "to the union of the statements in mathlib-train, synth-ineq", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 233, 505, 245 ], "spans": [ { "bbox": [ 106, 233, 288, 245 ], "score": 1.0, "content": "and miniF2F-curriculum. We uniformly set", "type": "text" }, { "bbox": [ 288, 234, 314, 244 ], "score": 0.89, "content": "a = 1", "type": "inline_equation" }, { "bbox": [ 314, 233, 465, 245 ], "score": 1.0, "content": "on mathlib-train and synth-ineq and", "type": "text" }, { "bbox": [ 465, 234, 491, 244 ], "score": 0.89, "content": "a = 8", "type": "inline_equation" }, { "bbox": [ 492, 233, 505, 245 ], "score": 1.0, "content": "on", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 245, 368, 255 ], "spans": [ { "bbox": [ 106, 245, 130, 255 ], "score": 1.0, "content": "miniF", "type": "text" }, { "bbox": [ 130, 245, 143, 254 ], "score": 0.61, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 143, 245, 223, 255 ], "score": 1.0, "content": "-curriculum and use", "type": "text" }, { "bbox": [ 224, 245, 234, 255 ], "score": 0.88, "content": "\\theta _ { 0 }", "type": "inline_equation" }, { "bbox": [ 235, 245, 252, 255 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 253, 245, 263, 255 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 263, 245, 368, 255 ], "score": 1.0, "content": "as described in Section 5.", "type": "text" } ], "index": 11 } ], "index": 10, "bbox_fs": [ 104, 221, 506, 255 ] }, { "type": "text", "bbox": [ 107, 261, 505, 327 ], "lines": [ { "bbox": [ 106, 261, 505, 273 ], "spans": [ { "bbox": [ 106, 261, 469, 273 ], "score": 1.0, "content": "Similarly to previous sections, we report in Figure 4 (left) the cumulative pass rate on miniF", "type": "text" }, { "bbox": [ 469, 262, 481, 271 ], "score": 0.43, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 482, 261, 505, 273 ], "score": 1.0, "content": "-valid", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 273, 505, 284 ], "spans": [ { "bbox": [ 106, 273, 505, 284 ], "score": 1.0, "content": "of our full curriculum expert iteration loop and compare them with the mathlib-train only expert", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 283, 506, 296 ], "spans": [ { "bbox": [ 105, 283, 506, 296 ], "score": 1.0, "content": "iteration from Section 4.5. Since more compute is deployed in our full-curriculum loop (more", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 294, 506, 307 ], "spans": [ { "bbox": [ 105, 294, 343, 307 ], "score": 1.0, "content": "statements), we also report a mathlib-train only loop taking", "type": "text" }, { "bbox": [ 344, 294, 369, 304 ], "score": 0.9, "content": "a = 2", "type": "inline_equation" }, { "bbox": [ 369, 294, 506, 307 ], "score": 1.0, "content": ". At the end of the expert iteration,", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 304, 506, 318 ], "spans": [ { "bbox": [ 105, 304, 273, 318 ], "score": 1.0, "content": "100 out of the 327 statements from miniF", "type": "text" }, { "bbox": [ 274, 306, 287, 315 ], "score": 0.32, "content": "2 F .", "type": "inline_equation" }, { "bbox": [ 287, 304, 506, 318 ], "score": 1.0, "content": "-curriculum end up being closed, suggesting a lack of", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 316, 316, 328 ], "spans": [ { "bbox": [ 106, 316, 316, 328 ], "score": 1.0, "content": "density in our manually formalized set of statement.", "type": "text" } ], "index": 17 } ], "index": 14.5, "bbox_fs": [ 105, 261, 506, 328 ] }, { "type": "text", "bbox": [ 106, 333, 506, 402 ], "lines": [ { "bbox": [ 105, 332, 506, 346 ], "spans": [ { "bbox": [ 105, 332, 269, 346 ], "score": 1.0, "content": "We also report in Figure 4 (right) the pass", "type": "text" }, { "bbox": [ 269, 333, 283, 343 ], "score": 0.69, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 283, 332, 318, 346 ], "score": 1.0, "content": "and pass", "type": "text" }, { "bbox": [ 319, 333, 333, 343 ], "score": 0.63, "content": "@ 8", "type": "inline_equation" }, { "bbox": [ 334, 332, 506, 346 ], "score": 1.0, "content": "for our full curriculum expert iteration loop.", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 344, 505, 356 ], "spans": [ { "bbox": [ 106, 344, 505, 356 ], "score": 1.0, "content": "The steady improvement on miniF2F-valid shows that the expert iteration procedure we propose does", "type": "text" } ], "index": 19 }, { "bbox": [ 104, 353, 506, 369 ], "spans": [ { "bbox": [ 104, 353, 506, 369 ], "score": 1.0, "content": "not overfit on the statements that compose the curriculum it uses. Despite the potential inefficiency", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 366, 506, 378 ], "spans": [ { "bbox": [ 106, 366, 506, 378 ], "score": 1.0, "content": "of our curriculum, the improved performance associated with its use demonstrates, as hypothesized,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 376, 506, 390 ], "spans": [ { "bbox": [ 105, 376, 251, 390 ], "score": 1.0, "content": "an effective transfer between miniF", "type": "text" }, { "bbox": [ 251, 377, 265, 387 ], "score": 0.43, "content": "{ } ^ { 7 2 F }", "type": "inline_equation" }, { "bbox": [ 265, 376, 506, 390 ], "score": 1.0, "content": "-curriculum, synth-ineq and miniF2F-valid through expert", "type": "text" } ], "index": 22 }, { "bbox": [ 103, 387, 404, 404 ], "spans": [ { "bbox": [ 103, 387, 384, 404 ], "score": 1.0, "content": "iteration. We will denote the fully iterated model from this section as", "type": "text" }, { "bbox": [ 384, 387, 400, 402 ], "score": 0.91, "content": "\\theta _ { 9 } ^ { f u l l }", "type": "inline_equation" }, { "bbox": [ 400, 387, 404, 404 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 23 } ], "index": 20.5, "bbox_fs": [ 103, 332, 506, 404 ] }, { "type": "image", "bbox": [ 123, 414, 489, 516 ], "blocks": [ { "type": "image_body", "bbox": [ 123, 414, 489, 516 ], "group_id": 0, "lines": [ { "bbox": [ 123, 414, 489, 516 ], "spans": [ { "bbox": [ 123, 414, 489, 516 ], "score": 0.958, "type": "image", "image_path": "04a8b08c5b09b13442660861e017e088784c96f4097bff3f86db3eb43c8ba58a.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 123, 414, 489, 448.0 ], "spans": [], "index": 24 }, { "bbox": [ 123, 448.0, 489, 482.0 ], "spans": [], "index": 25 }, { "bbox": [ 123, 482.0, 489, 516.0 ], "spans": [], "index": 26 } ] }, { "type": "image_caption", "bbox": [ 106, 531, 506, 619 ], "group_id": 0, "lines": [ { "bbox": [ 106, 531, 506, 544 ], "spans": [ { "bbox": [ 106, 531, 294, 544 ], "score": 1.0, "content": "Figure 4: Left: cumulative pass rate on miniF", "type": "text" }, { "bbox": [ 294, 532, 307, 541 ], "score": 0.26, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 307, 531, 506, 544 ], "score": 1.0, "content": "-valid for our expert iteration loop using our full", "type": "text" } ], "index": 27 }, { "bbox": [ 104, 541, 506, 555 ], "spans": [ { "bbox": [ 104, 541, 294, 555 ], "score": 1.0, "content": "curriculum (mathlib-train, synth-ineq and miniF", "type": "text" }, { "bbox": [ 295, 542, 307, 552 ], "score": 0.47, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 308, 541, 506, 555 ], "score": 1.0, "content": "-curriculum) compared to the expert iteration loop", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 552, 506, 565 ], "spans": [ { "bbox": [ 105, 552, 410, 565 ], "score": 1.0, "content": "from Section 4.5. The total number of attempts per iteration in our full loop is", "type": "text" }, { "bbox": [ 410, 553, 506, 564 ], "score": 0.9, "content": "2 5 k + 5 . 6 k + 8 * 3 2 7 \\approx", "type": "inline_equation" } ], "index": 29 }, { "bbox": [ 106, 564, 506, 576 ], "spans": [ { "bbox": [ 106, 564, 131, 574 ], "score": 0.85, "content": "3 3 . 2 k", "type": "inline_equation" }, { "bbox": [ 131, 564, 480, 576 ], "score": 1.0, "content": ", which means the total compute deployed is higher than in the mathlib-train only loop", "type": "text" }, { "bbox": [ 480, 564, 502, 575 ], "score": 0.81, "content": "( 2 5 k )", "type": "inline_equation" }, { "bbox": [ 503, 564, 506, 576 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 575, 506, 588 ], "spans": [ { "bbox": [ 105, 575, 381, 588 ], "score": 1.0, "content": "We therefore also report in dotted a mathlib-train only loop, taking", "type": "text" }, { "bbox": [ 382, 576, 407, 585 ], "score": 0.88, "content": "a = 2", "type": "inline_equation" }, { "bbox": [ 407, 575, 506, 588 ], "score": 1.0, "content": ", whose total number of", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 586, 505, 598 ], "spans": [ { "bbox": [ 105, 586, 202, 598 ], "score": 1.0, "content": "attempts per iteration is", "type": "text" }, { "bbox": [ 202, 586, 230, 596 ], "score": 0.88, "content": "\\approx 5 0 k", "type": "inline_equation" }, { "bbox": [ 230, 586, 283, 598 ], "score": 1.0, "content": ". Right: pass", "type": "text" }, { "bbox": [ 283, 586, 297, 596 ], "score": 0.45, "content": "@ l", "type": "inline_equation" }, { "bbox": [ 298, 586, 363, 598 ], "score": 1.0, "content": "(plain) and pass", "type": "text" }, { "bbox": [ 363, 586, 378, 596 ], "score": 0.61, "content": "@ 8", "type": "inline_equation" }, { "bbox": [ 378, 586, 505, 598 ], "score": 1.0, "content": "(dotted) for our expert iteration", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 597, 506, 610 ], "spans": [ { "bbox": [ 104, 597, 374, 610 ], "score": 1.0, "content": "loop using our full curriculum (mathlib-train, synth-ineq and miniF", "type": "text" }, { "bbox": [ 375, 597, 387, 607 ], "score": 0.36, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 387, 597, 506, 610 ], "score": 1.0, "content": "-curriculum) compared to the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 608, 262, 620 ], "spans": [ { "bbox": [ 105, 608, 262, 620 ], "score": 1.0, "content": "expert iteration loop from Section 4.5.", "type": "text" } ], "index": 34 } ], "index": 30.5 } ], "index": 27.75 }, { "type": "title", "bbox": [ 107, 637, 170, 649 ], "lines": [ { "bbox": [ 106, 636, 171, 650 ], "spans": [ { "bbox": [ 106, 636, 171, 650 ], "score": 1.0, "content": "6.2 RESULTS", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 107, 658, 505, 704 ], "lines": [ { "bbox": [ 105, 658, 505, 671 ], "spans": [ { "bbox": [ 105, 658, 505, 671 ], "score": 1.0, "content": "We report in Table 2 the pass rates on mathlib-{valid, test} and miniF2F-{valid, test} for the models", "type": "text" } ], "index": 36 }, { "bbox": [ 103, 666, 505, 687 ], "spans": [ { "bbox": [ 103, 666, 248, 687 ], "score": 1.0, "content": "trained in previous sections, namely", "type": "text" }, { "bbox": [ 248, 671, 259, 682 ], "score": 0.82, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 259, 666, 310, 687 ], "score": 1.0, "content": ", θmathlib9 , and", "type": "text" }, { "bbox": [ 310, 669, 326, 683 ], "score": 0.9, "content": "\\theta _ { 9 } ^ { f u l l }", "type": "inline_equation" }, { "bbox": [ 326, 666, 383, 687 ], "score": 1.0, "content": ". We achieve a", "type": "text" }, { "bbox": [ 384, 670, 411, 682 ], "score": 0.87, "content": "4 7 . 3 \\%", "type": "inline_equation" }, { "bbox": [ 411, 666, 474, 687 ], "score": 1.0, "content": "pass rate (using", "type": "text" }, { "bbox": [ 474, 671, 505, 681 ], "score": 0.87, "content": "a = 6 4", "type": "inline_equation" } ], "index": 37 }, { "bbox": [ 105, 681, 505, 695 ], "spans": [ { "bbox": [ 105, 681, 181, 695 ], "score": 1.0, "content": "attempts) on miniF", "type": "text" }, { "bbox": [ 181, 682, 194, 692 ], "score": 0.62, "content": "2 F .", "type": "inline_equation" }, { "bbox": [ 194, 681, 241, 695 ], "score": 1.0, "content": "-valid and a", "type": "text" }, { "bbox": [ 241, 682, 268, 692 ], "score": 0.87, "content": "3 6 . 6 \\%", "type": "inline_equation" }, { "bbox": [ 268, 681, 505, 695 ], "score": 1.0, "content": "pass rate on miniF2F-test, substantially improving from the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 693, 288, 705 ], "spans": [ { "bbox": [ 105, 693, 288, 705 ], "score": 1.0, "content": "previous state-of-the-art (Zheng et al., 2022).", "type": "text" } ], "index": 39 } ], "index": 37.5, "bbox_fs": [ 103, 658, 505, 705 ] }, { "type": "text", "bbox": [ 107, 709, 503, 732 ], "lines": [ { "bbox": [ 106, 710, 505, 721 ], "spans": [ { "bbox": [ 106, 710, 505, 721 ], "score": 1.0, "content": "These results include the resolution of 26 AMC12 problems, 6 AIME problems and 2 IMO-adapted", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "problems. Out of these statements, 4 AMC12 problems (amc12b_2020_p5, amc12a_2009_p9,", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 277, 505, 289 ], "spans": [ { "bbox": [ 106, 277, 505, 289 ], "score": 1.0, "content": "amc12a_2003_p24, amc12b_2003_p17), 2 AIME problems (aime_1984_p1, aime_1990_p4), and", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 105, 288, 505, 300 ], "spans": [ { "bbox": [ 105, 288, 215, 300 ], "score": 1.0, "content": "2 IMO-adapted problems", "type": "text", "cross_page": true }, { "bbox": [ 215, 288, 277, 300 ], "score": 0.57, "content": "( \\mathrm { i } \\mathsf { m o } _ { - } 1 9 6 1 \\mathsf { \\Pi } _ { - } \\mathsf { p } 1 ^ { 2 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 278, 288, 505, 300 ], "score": 1.0, "content": ", imo_1964_p2) are uniquely solved by expert iterated", "type": "text", "cross_page": true } ], "index": 7 }, { "bbox": [ 102, 293, 470, 318 ], "spans": [ { "bbox": [ 102, 293, 447, 318 ], "score": 1.0, "content": "models, the two IMO-adapted and the two AIME problems being uniquely solved by", "type": "text", "cross_page": true }, { "bbox": [ 448, 299, 464, 313 ], "score": 0.91, "content": "\\theta _ { 9 } ^ { \\bar { f } u l l }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 464, 293, 470, 318 ], "score": 1.0, "content": ".", "type": "text", "cross_page": true } ], "index": 8 } ], "index": 40.5, "bbox_fs": [ 105, 710, 505, 734 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 120, 125, 491, 254 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 80, 506, 117 ], "group_id": 0, "lines": [ { "bbox": [ 104, 79, 507, 94 ], "spans": [ { "bbox": [ 104, 79, 206, 94 ], "score": 1.0, "content": "Table 2: Performance of", "type": "text" }, { "bbox": [ 206, 81, 217, 92 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 217, 79, 341, 94 ], "score": 1.0, "content": "(value-function based search),", "type": "text" }, { "bbox": [ 342, 80, 369, 93 ], "score": 0.64, "content": "\\theta _ { 9 } ^ { m a t h l i b }", "type": "inline_equation" }, { "bbox": [ 370, 79, 507, 94 ], "score": 1.0, "content": "(expert iterated on mathlib-train)", "type": "text" } ], "index": 0 }, { "bbox": [ 102, 91, 507, 108 ], "spans": [ { "bbox": [ 102, 91, 123, 108 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 92, 140, 106 ], "score": 0.91, "content": "\\theta _ { 9 } ^ { f u l l }", "type": "inline_equation" }, { "bbox": [ 141, 91, 507, 108 ], "score": 1.0, "content": "(expert iterated on our full curriculum) on mathlib-{valid, test} and miniF2F-{valid, test}.", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 105, 313, 117 ], "spans": [ { "bbox": [ 105, 105, 232, 117 ], "score": 1.0, "content": "All proof searches are run with", "type": "text" }, { "bbox": [ 232, 105, 267, 115 ], "score": 0.9, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 268, 105, 285, 117 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 285, 106, 309, 115 ], "score": 0.88, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 309, 105, 313, 117 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 120, 125, 491, 254 ], "group_id": 0, "lines": [ { "bbox": [ 120, 125, 491, 254 ], "spans": [ { "bbox": [ 120, 125, 491, 254 ], "score": 0.983, "html": "
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", "type": "table", "image_path": "24b26020e52517392c633e317f9316987df86384fa0246f8d4d22e1a4ba7a0e4.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 120, 125, 491, 168.0 ], "spans": [], "index": 3 }, { "bbox": [ 120, 168.0, 491, 211.0 ], "spans": [], "index": 4 }, { "bbox": [ 120, 211.0, 491, 254.0 ], "spans": [], "index": 5 } ] } ], "index": 2.5 }, { "type": "text", "bbox": [ 107, 277, 505, 312 ], "lines": [ { "bbox": [ 106, 277, 505, 289 ], "spans": [ { "bbox": [ 106, 277, 505, 289 ], "score": 1.0, "content": "amc12a_2003_p24, amc12b_2003_p17), 2 AIME problems (aime_1984_p1, aime_1990_p4), and", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 288, 505, 300 ], "spans": [ { "bbox": [ 105, 288, 215, 300 ], "score": 1.0, "content": "2 IMO-adapted problems", "type": "text" }, { "bbox": [ 215, 288, 277, 300 ], "score": 0.57, "content": "( \\mathrm { i } \\mathsf { m o } _ { - } 1 9 6 1 \\mathsf { \\Pi } _ { - } \\mathsf { p } 1 ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 278, 288, 505, 300 ], "score": 1.0, "content": ", imo_1964_p2) are uniquely solved by expert iterated", "type": "text" } ], "index": 7 }, { "bbox": [ 102, 293, 470, 318 ], "spans": [ { "bbox": [ 102, 293, 447, 318 ], "score": 1.0, "content": "models, the two IMO-adapted and the two AIME problems being uniquely solved by", "type": "text" }, { "bbox": [ 448, 299, 464, 313 ], "score": 0.91, "content": "\\theta _ { 9 } ^ { \\bar { f } u l l }", "type": "inline_equation" }, { "bbox": [ 464, 293, 470, 318 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 8 } ], "index": 7 }, { "type": "text", "bbox": [ 107, 317, 505, 362 ], "lines": [ { "bbox": [ 106, 318, 505, 330 ], "spans": [ { "bbox": [ 106, 318, 505, 330 ], "score": 1.0, "content": "We provide a selection of the proofs found by our models for these statements as well as a qualitative", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 329, 505, 340 ], "spans": [ { "bbox": [ 106, 330, 447, 340 ], "score": 1.0, "content": "analysis of them in Appendix J. Also, we achieve a new state-of-the-art: higher than", "type": "text" }, { "bbox": [ 447, 329, 467, 340 ], "score": 0.88, "content": "7 5 \\%", "type": "inline_equation" }, { "bbox": [ 467, 330, 505, 340 ], "score": 1.0, "content": "pass rate", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 340, 505, 352 ], "spans": [ { "bbox": [ 105, 340, 134, 352 ], "score": 1.0, "content": "(using", "type": "text" }, { "bbox": [ 134, 340, 166, 351 ], "score": 0.89, "content": "a = 6 4", "type": "inline_equation" }, { "bbox": [ 166, 340, 505, 352 ], "score": 1.0, "content": "attempts) on mathlib-{valid, test}, suggesting that our models could potentially be", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 351, 475, 363 ], "spans": [ { "bbox": [ 106, 351, 475, 363 ], "score": 1.0, "content": "effectively leveraged as proof assistants in the formalization efforts associated with mathlib.", "type": "text" } ], "index": 12 } ], "index": 10.5 }, { "type": "title", "bbox": [ 109, 378, 280, 391 ], "lines": [ { "bbox": [ 105, 378, 282, 393 ], "spans": [ { "bbox": [ 105, 378, 282, 393 ], "score": 1.0, "content": "7 DISCUSSION AND LIMITATION", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 107, 403, 505, 502 ], "lines": [ { "bbox": [ 106, 402, 505, 416 ], "spans": [ { "bbox": [ 106, 402, 319, 416 ], "score": 1.0, "content": "Throughout this paper, we used a single model size (", "type": "text" }, { "bbox": [ 320, 403, 344, 414 ], "score": 0.36, "content": "7 7 4 \\mathrm { m }", "type": "inline_equation" }, { "bbox": [ 344, 402, 505, 416 ], "score": 1.0, "content": "trainable parameters). We refer readers", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 414, 506, 428 ], "spans": [ { "bbox": [ 105, 414, 506, 428 ], "score": 1.0, "content": "to Appendix H for more discussion on model size, compute budget and training time. Despite our", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 426, 505, 438 ], "spans": [ { "bbox": [ 106, 426, 505, 438 ], "score": 1.0, "content": "models’ capability, as discussed in Appendix J.1, to generate cuts and witnesses, we believe that their", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 436, 506, 449 ], "spans": [ { "bbox": [ 105, 436, 506, 449 ], "score": 1.0, "content": "current main limitation lies in their inability (under our proposed search procedure) to chain more", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 446, 505, 461 ], "spans": [ { "bbox": [ 105, 446, 505, 461 ], "score": 1.0, "content": "than 2 or 3 non-trivial steps of mathematical reasoning, preventing them from consistently solving", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 458, 506, 471 ], "spans": [ { "bbox": [ 105, 458, 506, 471 ], "score": 1.0, "content": "challenging olympiad problems. We’ve been repeatedly impressed by the complexity of some of", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 469, 505, 482 ], "spans": [ { "bbox": [ 105, 469, 505, 482 ], "score": 1.0, "content": "the proofsteps generated by our models. But, proofs requiring many of such reasoning steps remain", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 480, 506, 493 ], "spans": [ { "bbox": [ 105, 480, 506, 493 ], "score": 1.0, "content": "beyond our current compute horizon. Even if we solved a selection of challenging olympiad problems,", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 492, 491, 504 ], "spans": [ { "bbox": [ 106, 492, 491, 504 ], "score": 1.0, "content": "our models are still far from being competitive with the brightest students in these competitions.", "type": "text" } ], "index": 22 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 508, 505, 574 ], "lines": [ { "bbox": [ 105, 507, 505, 521 ], "spans": [ { "bbox": [ 105, 507, 505, 521 ], "score": 1.0, "content": "While our models have demonstrated some capabilities to generate cuts, the cuts they generate are", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 518, 505, 532 ], "spans": [ { "bbox": [ 105, 518, 505, 532 ], "score": 1.0, "content": "often shallow (they involve only a few proofsteps and don’t necessarily deeply change the structure", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 529, 506, 542 ], "spans": [ { "bbox": [ 105, 529, 506, 542 ], "score": 1.0, "content": "of the proof–we refer the reader to the Cut-Elimination theorem and Carbone & Semmes (1996) for a", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 539, 505, 554 ], "spans": [ { "bbox": [ 105, 539, 505, 554 ], "score": 1.0, "content": "discussion of the influence of cuts on proof size). 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(2021)), are interesting avenues of research to alleviate this limitation.", "type": "text" } ], "index": 28 } ], "index": 25.5 }, { "type": "title", "bbox": [ 108, 590, 195, 603 ], "lines": [ { "bbox": [ 104, 587, 198, 606 ], "spans": [ { "bbox": [ 104, 587, 198, 606 ], "score": 1.0, "content": "8 CONCLUSION", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 615, 506, 703 ], "lines": [ { "bbox": [ 105, 615, 507, 628 ], "spans": [ { "bbox": [ 105, 615, 507, 628 ], "score": 1.0, "content": "In this paper we presented an expert iteration procedure for GPT-f (Polu & Sutskever, 2020), demon-", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 627, 506, 639 ], "spans": [ { "bbox": [ 106, 627, 506, 639 ], "score": 1.0, "content": "strating that it is capable of solving a curriculum of increasingly difficult problems out of a set of", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 637, 505, 650 ], "spans": [ { "bbox": [ 105, 637, 505, 650 ], "score": 1.0, "content": "formal statements of sufficiently varied difficulty. Our results suggest that the lack of self-play in", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 648, 506, 660 ], "spans": [ { "bbox": [ 105, 648, 506, 660 ], "score": 1.0, "content": "the formal mathematics setup can be effectively compensated for by automatically/manually curated", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 658, 506, 673 ], "spans": [ { "bbox": [ 105, 658, 506, 673 ], "score": 1.0, "content": "sets of formal statements, which are much cheaper to formalize than full proofs. Finally, we hope", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 669, 506, 683 ], "spans": [ { "bbox": [ 105, 669, 506, 683 ], "score": 1.0, "content": "that the statement curriculum learning methodology we presented in this work will help accelerate", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 682, 506, 694 ], "spans": [ { "bbox": [ 106, 682, 506, 694 ], "score": 1.0, "content": "progress in automated reasoning, especially if scaled with automated generation and curation of", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 691, 234, 704 ], "spans": [ { "bbox": [ 106, 691, 234, 704 ], "score": 1.0, "content": "formal statements in the future.", "type": "text" } ], "index": 37 } ], "index": 33.5 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 711, 505, 731 ], "lines": [ { "bbox": [ 117, 709, 506, 725 ], "spans": [ { "bbox": [ 117, 709, 506, 725 ], "score": 1.0, "content": "2This IMO-adapted statement from miniF2F-valid is a much weaker version than the original problem (see", "type": "text" } ] }, { "bbox": [ 106, 721, 217, 733 ], "spans": [ { "bbox": [ 106, 721, 217, 733 ], "score": 1.0, "content": "Appendix J for more context).", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 25, 293, 38 ], "spans": [ { "bbox": [ 106, 25, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 759 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "9", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 120, 125, 491, 254 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 80, 506, 117 ], "group_id": 0, "lines": [ { "bbox": [ 104, 79, 507, 94 ], "spans": [ { "bbox": [ 104, 79, 206, 94 ], "score": 1.0, "content": "Table 2: Performance of", "type": "text" }, { "bbox": [ 206, 81, 217, 92 ], "score": 0.88, "content": "\\theta _ { 1 }", "type": "inline_equation" }, { "bbox": [ 217, 79, 341, 94 ], "score": 1.0, "content": "(value-function based search),", "type": "text" }, { "bbox": [ 342, 80, 369, 93 ], "score": 0.64, "content": "\\theta _ { 9 } ^ { m a t h l i b }", "type": "inline_equation" }, { "bbox": [ 370, 79, 507, 94 ], "score": 1.0, "content": "(expert iterated on mathlib-train)", "type": "text" } ], "index": 0 }, { "bbox": [ 102, 91, 507, 108 ], "spans": [ { "bbox": [ 102, 91, 123, 108 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 92, 140, 106 ], "score": 0.91, "content": "\\theta _ { 9 } ^ { f u l l }", "type": "inline_equation" }, { "bbox": [ 141, 91, 507, 108 ], "score": 1.0, "content": "(expert iterated on our full curriculum) on mathlib-{valid, test} and miniF2F-{valid, test}.", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 105, 313, 117 ], "spans": [ { "bbox": [ 105, 105, 232, 117 ], "score": 1.0, "content": "All proof searches are run with", "type": "text" }, { "bbox": [ 232, 105, 267, 115 ], "score": 0.9, "content": "d = 5 1 2", "type": "inline_equation" }, { "bbox": [ 268, 105, 285, 117 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 285, 106, 309, 115 ], "score": 0.88, "content": "e = 8", "type": "inline_equation" }, { "bbox": [ 309, 105, 313, 117 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 120, 125, 491, 254 ], "group_id": 0, "lines": [ { "bbox": [ 120, 125, 491, 254 ], "spans": [ { "bbox": [ 120, 125, 491, 254 ], "score": 0.983, "html": "
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", "type": "table", "image_path": "24b26020e52517392c633e317f9316987df86384fa0246f8d4d22e1a4ba7a0e4.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 120, 125, 491, 168.0 ], "spans": [], "index": 3 }, { "bbox": [ 120, 168.0, 491, 211.0 ], "spans": [], "index": 4 }, { "bbox": [ 120, 211.0, 491, 254.0 ], "spans": [], "index": 5 } ] } ], "index": 2.5 }, { "type": "text", "bbox": [ 107, 277, 505, 312 ], "lines": [], "index": 7, "bbox_fs": [ 102, 277, 505, 318 ], "lines_deleted": true }, { "type": "text", "bbox": [ 107, 317, 505, 362 ], "lines": [ { "bbox": [ 106, 318, 505, 330 ], "spans": [ { "bbox": [ 106, 318, 505, 330 ], "score": 1.0, "content": "We provide a selection of the proofs found by our models for these statements as well as a qualitative", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 329, 505, 340 ], "spans": [ { "bbox": [ 106, 330, 447, 340 ], "score": 1.0, "content": "analysis of them in Appendix J. Also, we achieve a new state-of-the-art: higher than", "type": "text" }, { "bbox": [ 447, 329, 467, 340 ], "score": 0.88, "content": "7 5 \\%", "type": "inline_equation" }, { "bbox": [ 467, 330, 505, 340 ], "score": 1.0, "content": "pass rate", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 340, 505, 352 ], "spans": [ { "bbox": [ 105, 340, 134, 352 ], "score": 1.0, "content": "(using", "type": "text" }, { "bbox": [ 134, 340, 166, 351 ], "score": 0.89, "content": "a = 6 4", "type": "inline_equation" }, { "bbox": [ 166, 340, 505, 352 ], "score": 1.0, "content": "attempts) on mathlib-{valid, test}, suggesting that our models could potentially be", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 351, 475, 363 ], "spans": [ { "bbox": [ 106, 351, 475, 363 ], "score": 1.0, "content": "effectively leveraged as proof assistants in the formalization efforts associated with mathlib.", "type": "text" } ], "index": 12 } ], "index": 10.5, "bbox_fs": [ 105, 318, 505, 363 ] }, { "type": "title", "bbox": [ 109, 378, 280, 391 ], "lines": [ { "bbox": [ 105, 378, 282, 393 ], "spans": [ { "bbox": [ 105, 378, 282, 393 ], "score": 1.0, "content": "7 DISCUSSION AND LIMITATION", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 107, 403, 505, 502 ], "lines": [ { "bbox": [ 106, 402, 505, 416 ], "spans": [ { "bbox": [ 106, 402, 319, 416 ], "score": 1.0, "content": "Throughout this paper, we used a single model size (", "type": "text" }, { "bbox": [ 320, 403, 344, 414 ], "score": 0.36, "content": "7 7 4 \\mathrm { m }", "type": "inline_equation" }, { "bbox": [ 344, 402, 505, 416 ], "score": 1.0, "content": "trainable parameters). We refer readers", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 414, 506, 428 ], "spans": [ { "bbox": [ 105, 414, 506, 428 ], "score": 1.0, "content": "to Appendix H for more discussion on model size, compute budget and training time. Despite our", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 426, 505, 438 ], "spans": [ { "bbox": [ 106, 426, 505, 438 ], "score": 1.0, "content": "models’ capability, as discussed in Appendix J.1, to generate cuts and witnesses, we believe that their", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 436, 506, 449 ], "spans": [ { "bbox": [ 105, 436, 506, 449 ], "score": 1.0, "content": "current main limitation lies in their inability (under our proposed search procedure) to chain more", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 446, 505, 461 ], "spans": [ { "bbox": [ 105, 446, 505, 461 ], "score": 1.0, "content": "than 2 or 3 non-trivial steps of mathematical reasoning, preventing them from consistently solving", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 458, 506, 471 ], "spans": [ { "bbox": [ 105, 458, 506, 471 ], "score": 1.0, "content": "challenging olympiad problems. We’ve been repeatedly impressed by the complexity of some of", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 469, 505, 482 ], "spans": [ { "bbox": [ 105, 469, 505, 482 ], "score": 1.0, "content": "the proofsteps generated by our models. But, proofs requiring many of such reasoning steps remain", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 480, 506, 493 ], "spans": [ { "bbox": [ 105, 480, 506, 493 ], "score": 1.0, "content": "beyond our current compute horizon. Even if we solved a selection of challenging olympiad problems,", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 492, 491, 504 ], "spans": [ { "bbox": [ 106, 492, 491, 504 ], "score": 1.0, "content": "our models are still far from being competitive with the brightest students in these competitions.", "type": "text" } ], "index": 22 } ], "index": 18, "bbox_fs": [ 105, 402, 506, 504 ] }, { "type": "text", "bbox": [ 107, 508, 505, 574 ], "lines": [ { "bbox": [ 105, 507, 505, 521 ], "spans": [ { "bbox": [ 105, 507, 505, 521 ], "score": 1.0, "content": "While our models have demonstrated some capabilities to generate cuts, the cuts they generate are", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 518, 505, 532 ], "spans": [ { "bbox": [ 105, 518, 505, 532 ], "score": 1.0, "content": "often shallow (they involve only a few proofsteps and don’t necessarily deeply change the structure", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 529, 506, 542 ], "spans": [ { "bbox": [ 105, 529, 506, 542 ], "score": 1.0, "content": "of the proof–we refer the reader to the Cut-Elimination theorem and Carbone & Semmes (1996) for a", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 539, 505, 554 ], "spans": [ { "bbox": [ 105, 539, 505, 554 ], "score": 1.0, "content": "discussion of the influence of cuts on proof size). We believe that studying language models’ ability", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 552, 505, 564 ], "spans": [ { "bbox": [ 105, 552, 505, 564 ], "score": 1.0, "content": "to generate cuts, and designing search procedures that leverage that capability (related ideas can be", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 563, 496, 575 ], "spans": [ { "bbox": [ 106, 563, 496, 575 ], "score": 1.0, "content": "found in Czechowski et al. (2021)), are interesting avenues of research to alleviate this limitation.", "type": "text" } ], "index": 28 } ], "index": 25.5, "bbox_fs": [ 105, 507, 506, 575 ] }, { "type": "title", "bbox": [ 108, 590, 195, 603 ], "lines": [ { "bbox": [ 104, 587, 198, 606 ], "spans": [ { "bbox": [ 104, 587, 198, 606 ], "score": 1.0, "content": "8 CONCLUSION", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 615, 506, 703 ], "lines": [ { "bbox": [ 105, 615, 507, 628 ], "spans": [ { "bbox": [ 105, 615, 507, 628 ], "score": 1.0, "content": "In this paper we presented an expert iteration procedure for GPT-f (Polu & Sutskever, 2020), demon-", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 627, 506, 639 ], "spans": [ { "bbox": [ 106, 627, 506, 639 ], "score": 1.0, "content": "strating that it is capable of solving a curriculum of increasingly difficult problems out of a set of", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 637, 505, 650 ], "spans": [ { "bbox": [ 105, 637, 505, 650 ], "score": 1.0, "content": "formal statements of sufficiently varied difficulty. Our results suggest that the lack of self-play in", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 648, 506, 660 ], "spans": [ { "bbox": [ 105, 648, 506, 660 ], "score": 1.0, "content": "the formal mathematics setup can be effectively compensated for by automatically/manually curated", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 658, 506, 673 ], "spans": [ { "bbox": [ 105, 658, 506, 673 ], "score": 1.0, "content": "sets of formal statements, which are much cheaper to formalize than full proofs. Finally, we hope", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 669, 506, 683 ], "spans": [ { "bbox": [ 105, 669, 506, 683 ], "score": 1.0, "content": "that the statement curriculum learning methodology we presented in this work will help accelerate", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 682, 506, 694 ], "spans": [ { "bbox": [ 106, 682, 506, 694 ], "score": 1.0, "content": "progress in automated reasoning, especially if scaled with automated generation and curation of", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 691, 234, 704 ], "spans": [ { "bbox": [ 106, 691, 234, 704 ], "score": 1.0, "content": "formal statements in the future.", "type": "text" } ], "index": 37 } ], "index": 33.5, "bbox_fs": [ 105, 615, 507, 704 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 81, 176, 94 ], "lines": [ { "bbox": [ 106, 82, 176, 94 ], "spans": [ { "bbox": [ 106, 82, 176, 94 ], "score": 1.0, "content": "REFERENCES", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 99, 374, 111 ], "lines": [ { "bbox": [ 105, 99, 375, 113 ], "spans": [ { "bbox": [ 105, 99, 375, 113 ], "score": 1.0, "content": "Lean theorem prover. https://leanprover.github.io/about/.", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 107, 117, 504, 140 ], "lines": [ { "bbox": [ 105, 115, 505, 132 ], "spans": [ { "bbox": [ 105, 115, 505, 132 ], "score": 1.0, "content": "Kshitij Bansal, Sarah M Loos, Markus N Rabe, and Christian Szegedy. 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GamePad (Huang et al., 2019) and CoqGymn/ASTactic (Yang & Deng, 2019)", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 293, 505, 307 ], "spans": [ { "bbox": [ 105, 293, 505, 307 ], "score": 1.0, "content": "introduce environments based on the Coq theorem prover. ASTactic generates tactics as programs by", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 305, 505, 317 ], "spans": [ { "bbox": [ 106, 305, 505, 317 ], "score": 1.0, "content": "sequentially expanding a partial abstract syntax tree. Urban & Jakubuv (2020) studied the capability", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 315, 505, 328 ], "spans": [ { "bbox": [ 106, 315, 505, 328 ], "score": 1.0, "content": "of GPT-2 to produce useful conjectures for the Mizar library and IsarStep (Li et al., 2021) explored", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 327, 506, 339 ], "spans": [ { "bbox": [ 106, 327, 506, 339 ], "score": 1.0, "content": "the synthesis of intermediate propositions in declarative proofs for Isabelle/HOL using Transformers.", "type": "text" } ], "index": 20 } ], "index": 15.5, "bbox_fs": [ 104, 227, 506, 339 ] }, { "type": "text", "bbox": [ 107, 350, 505, 427 ], "lines": [ { "bbox": [ 106, 349, 506, 363 ], "spans": [ { "bbox": [ 106, 349, 506, 363 ], "score": 1.0, "content": "Targeting miniF2F Lample et al. (2022) designed HyperTree Proof Search (HTPS), an online", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 361, 505, 373 ], "spans": [ { "bbox": [ 105, 361, 505, 373 ], "score": 1.0, "content": "training procedure targeting Lean, Metamath and hand-crafted environment named Equations. Lample", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 371, 505, 384 ], "spans": [ { "bbox": [ 106, 371, 184, 384 ], "score": 1.0, "content": "et al. 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Thor (Jiang et al., 2022) combined language model and", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 392, 507, 407 ], "spans": [ { "bbox": [ 105, 392, 295, 407 ], "score": 1.0, "content": "Sledgehammer (Paulson, 2010) and achieved", "type": "text" }, { "bbox": [ 295, 394, 323, 404 ], "score": 0.87, "content": "2 9 . 9 \\%", "type": "inline_equation" }, { "bbox": [ 324, 392, 402, 407 ], "score": 1.0, "content": "pass-rate on miniF", "type": "text" }, { "bbox": [ 402, 394, 415, 404 ], "score": 0.29, "content": "2 F", "type": "inline_equation" }, { "bbox": [ 415, 392, 507, 407 ], "score": 1.0, "content": "-test in Isabelle setup,", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 404, 505, 417 ], "spans": [ { "bbox": [ 105, 404, 220, 417 ], "score": 1.0, "content": "which is later improved to", "type": "text" }, { "bbox": [ 220, 405, 248, 415 ], "score": 0.87, "content": "3 5 . 2 \\%", "type": "inline_equation" }, { "bbox": [ 249, 404, 263, 417 ], "score": 1.0, "content": "by", "type": "text" }, { "bbox": [ 264, 405, 280, 415 ], "score": 0.29, "content": "\\mathbf { W } \\mathbf { u }", "type": "inline_equation" }, { "bbox": [ 281, 404, 505, 417 ], "score": 1.0, "content": "et al. (2022) leveraging autoformalization and expert", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 415, 145, 428 ], "spans": [ { "bbox": [ 105, 415, 145, 428 ], "score": 1.0, "content": "iteration.", "type": "text" } ], "index": 27 } ], "index": 24, "bbox_fs": [ 105, 349, 507, 428 ] }, { "type": "title", "bbox": [ 108, 443, 186, 455 ], "lines": [ { "bbox": [ 105, 441, 188, 457 ], "spans": [ { "bbox": [ 105, 441, 188, 457 ], "score": 1.0, "content": "B LEAN-GYM", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 108, 467, 260, 479 ], "lines": [ { "bbox": [ 106, 467, 261, 480 ], "spans": [ { "bbox": [ 106, 467, 261, 480 ], "score": 1.0, "content": "lean-gym presents the following API:", "type": "text" } ], "index": 29 } ], "index": 29, "bbox_fs": [ 106, 467, 261, 480 ] }, { "type": "list", "bbox": [ 133, 488, 505, 569 ], "lines": [ { "bbox": [ 133, 488, 506, 501 ], "spans": [ { "bbox": [ 133, 488, 256, 501 ], "score": 1.0, "content": "• init-search: declaration", "type": "text" }, { "bbox": [ 256, 490, 270, 498 ], "score": 0.55, "content": "", "type": "inline_equation" }, { "bbox": [ 270, 488, 506, 501 ], "score": 1.0, "content": "tactic_state. 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It returns the initial tactic state along with a fresh search_id", "type": "text" } ], "index": 32 }, { "bbox": [ 141, 523, 239, 532 ], "spans": [ { "bbox": [ 141, 523, 239, 532 ], "score": 1.0, "content": "and tactic_state_id.", "type": "text" } ], "index": 33, "is_list_end_line": true }, { "bbox": [ 133, 536, 506, 547 ], "spans": [ { "bbox": [ 133, 536, 289, 547 ], "score": 1.0, "content": "• run_tac: (tactic_state, tactic)", "type": "text" }, { "bbox": [ 289, 537, 304, 547 ], "score": 0.7, "content": "", "type": "inline_equation" }, { "bbox": [ 305, 536, 372, 547 ], "score": 1.0, "content": "tactic_state.", "type": "text" }, { "bbox": [ 380, 536, 506, 547 ], "score": 1.0, "content": "Takes a search_id and a", "type": "text" } ], "index": 34, "is_list_start_line": true }, { "bbox": [ 142, 547, 506, 559 ], "spans": [ { "bbox": [ 142, 547, 506, 559 ], "score": 1.0, "content": "tactic_state_id to identify a tactic state, as well as a tactic string to apply to it. It", "type": "text" } ], "index": 35 }, { "bbox": [ 141, 559, 395, 570 ], "spans": [ { "bbox": [ 141, 559, 395, 570 ], "score": 1.0, "content": "returns a new tactic state and its associated tactic_state_id.", "type": "text" } ], "index": 36, "is_list_end_line": true } ], "index": 33, "bbox_fs": [ 133, 488, 506, 570 ] }, { "type": "text", "bbox": [ 103, 579, 475, 591 ], "lines": [ { "bbox": [ 105, 578, 477, 592 ], "spans": [ { "bbox": [ 105, 578, 477, 592 ], "score": 1.0, "content": "Below is an example in-terminal trace demonstrating the use of lean-gym’s REPL interface:", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 105, 578, 477, 592 ] }, { "type": "list", "bbox": [ 104, 600, 421, 732 ], "lines": [ { "bbox": [ 106, 600, 239, 612 ], "spans": [ { "bbox": [ 106, 600, 114, 610 ], "score": 0.63, "content": "\\$ 1", "type": "inline_equation" }, { "bbox": [ 114, 600, 239, 612 ], "score": 1.0, "content": "lean --run src/repl.lean", "type": "text" } ], "index": 38, 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Writing a wrapper in Python, as an example, only takes a few", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 219, 505, 232 ], "spans": [ { "bbox": [ 106, 219, 505, 232 ], "score": 1.0, "content": "dozen lines of code. Since lean-gym is a Lean program, managing the loaded libraries is done", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 230, 505, 244 ], "spans": [ { "bbox": [ 105, 230, 505, 244 ], "score": 1.0, "content": "directly using Lean’s own infrastructure (using leanpkg.toml), making it quite straightforward to", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 242, 457, 254 ], "spans": [ { "bbox": [ 105, 242, 457, 254 ], "score": 1.0, "content": "have access to both mathlib and miniF2F statements from the same lean-gym instance.", "type": "text" } ], "index": 12 } ], "index": 10 }, { "type": "text", "bbox": [ 106, 258, 505, 314 ], "lines": [ { "bbox": [ 105, 258, 506, 272 ], "spans": [ { "bbox": [ 105, 258, 506, 272 ], "score": 1.0, "content": "Note that lean-gym is stateful, meaning that distributing proof searches on multiple lean-gym", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 270, 505, 282 ], "spans": [ { "bbox": [ 105, 270, 505, 282 ], "score": 1.0, "content": "instances requires to track which instance is associated with which proof search. In practice, we were", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 281, 506, 293 ], "spans": [ { "bbox": [ 106, 281, 506, 293 ], "score": 1.0, "content": "able to scale the use of lean-gym to thousands of cores running thousands of proof searches in parallel.", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 290, 505, 304 ], "spans": [ { "bbox": [ 105, 290, 505, 304 ], "score": 1.0, "content": "Finally, lean-gym’s REPL interface is blocking, preventing inner-proof search parallelization, though", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 303, 321, 315 ], "spans": [ { "bbox": [ 106, 303, 321, 315 ], "score": 1.0, "content": "this limitation can probably be removed in the future.", "type": "text" } ], "index": 17 } ], "index": 15 }, { "type": "title", "bbox": [ 107, 330, 185, 343 ], "lines": [ { "bbox": [ 105, 327, 187, 345 ], "spans": [ { "bbox": [ 105, 327, 187, 345 ], "score": 1.0, "content": "C WEBMATH", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 106, 355, 437, 367 ], "lines": [ { "bbox": [ 106, 354, 438, 369 ], "spans": [ { "bbox": [ 106, 354, 438, 369 ], "score": 1.0, "content": "Our updated WebMath pre-training dataset consists in the mix presented in table 3.", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "table", "bbox": [ 225, 402, 385, 496 ], "blocks": [ { "type": "table_caption", "bbox": [ 146, 376, 463, 389 ], "group_id": 0, "lines": [ { "bbox": [ 145, 374, 465, 391 ], "spans": [ { "bbox": [ 145, 374, 465, 391 ], "score": 1.0, "content": "Table 3: Mix and source of data involved in the updated WebMath pre-training.", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "table_body", "bbox": [ 225, 402, 385, 496 ], "group_id": 0, "lines": [ { "bbox": [ 225, 402, 385, 496 ], "spans": [ { "bbox": [ 225, 402, 385, 496 ], "score": 0.968, "html": "
DatasetSizeMix
Github Python179 GB25%
arXiv Math10 GB25%
Math StackExchange2GB25%
PACT mix228GB17%
Math Overflow200 M5%
ProofWiki30M2%
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DatasetSizeMix
Github Python179 GB25%
arXiv Math10 GB25%
Math StackExchange2GB25%
PACT mix228GB17%
Math Overflow200 M5%
ProofWiki30M2%
PlanetMath25M1%
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Lean environment parses the statement and exposes to users the", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 433, 506, 446 ], "spans": [ { "bbox": [ 105, 433, 506, 446 ], "score": 1.0, "content": "goal to be proved. The model outputs a line of code (tactics and corresponding arguments). Lean", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 444, 506, 456 ], "spans": [ { "bbox": [ 106, 444, 506, 456 ], "score": 1.0, "content": "environment receives the model output and transforms the previous goal to another goal to be proved.", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 456, 505, 467 ], "spans": [ { "bbox": [ 106, 456, 505, 467 ], "score": 1.0, "content": "This process is repeated till all remaining goals are closed. In this case, the original statement is", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 466, 430, 479 ], "spans": [ { "bbox": [ 106, 466, 430, 479 ], "score": 1.0, "content": "proved: the final proof is collected by following the trajectory of model’s output.", "type": "text" } ], "index": 12 } ], "index": 9.5 } ], "index": 7.25 }, { "type": "title", "bbox": [ 106, 510, 323, 523 ], "lines": [ { "bbox": [ 105, 509, 324, 524 ], "spans": [ { "bbox": [ 105, 509, 324, 524 ], "score": 1.0, "content": "E ILLUSTRATION OF EXPERT ITERATION", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "image", "bbox": [ 109, 544, 502, 686 ], "blocks": [ { "type": "image_body", "bbox": [ 109, 544, 502, 686 ], "group_id": 1, "lines": [ { "bbox": [ 109, 544, 502, 686 ], "spans": [ { "bbox": [ 109, 544, 502, 686 ], "score": 0.968, "type": "image", "image_path": "9797ed05f1f348d3cda8cf3dcbcc103be5583e5290f598625053ea3f06dfab7b.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 109, 544, 502, 591.3333333333334 ], "spans": [], "index": 14 }, { "bbox": [ 109, 591.3333333333334, 502, 638.6666666666667 ], "spans": [], "index": 15 }, { "bbox": [ 109, 638.6666666666667, 502, 686.0000000000001 ], "spans": [], "index": 16 } ] }, { "type": "image_caption", "bbox": [ 103, 695, 505, 719 ], "group_id": 1, "lines": [ { "bbox": [ 105, 695, 505, 708 ], "spans": [ { "bbox": [ 105, 695, 505, 708 ], "score": 1.0, "content": "Figure 6: Illustration of expert iteration. The notation in this figure corresponds to Section 4.4 in", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 707, 149, 718 ], "spans": [ { "bbox": [ 105, 707, 149, 718 ], "score": 1.0, "content": "main text.", "type": "text" } ], "index": 18 } ], "index": 17.5 } ], "index": 16.25 } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 81, 262, 94 ], "lines": [ { "bbox": [ 105, 80, 263, 96 ], "spans": [ { "bbox": [ 105, 80, 263, 96 ], "score": 1.0, "content": "F SYNTHETIC INEQUALITIES", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 106, 165, 118 ], "lines": [ { "bbox": [ 105, 105, 166, 119 ], "spans": [ { "bbox": [ 105, 105, 166, 119 ], "score": 1.0, "content": "F.1 DESIGN", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 108, 127, 262, 138 ], "lines": [ { "bbox": [ 106, 127, 263, 140 ], "spans": [ { "bbox": [ 106, 127, 263, 140 ], "score": 1.0, "content": "The generator consists of three phases:", "type": "text" } ], "index": 2 } ], "index": 2 }, { "type": "text", "bbox": [ 107, 150, 505, 227 ], "lines": [ { "bbox": [ 106, 151, 505, 163 ], "spans": [ { "bbox": [ 106, 151, 505, 163 ], "score": 1.0, "content": "Seed expressions generation The first phase consists in generating seed expressions for which", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 161, 506, 174 ], "spans": [ { "bbox": [ 105, 161, 351, 174 ], "score": 1.0, "content": "we track the sign. We start by initializing an expression set", "type": "text" }, { "bbox": [ 351, 162, 360, 171 ], "score": 0.8, "content": "E", "type": "inline_equation" }, { "bbox": [ 361, 161, 506, 174 ], "score": 1.0, "content": "composed of tuples of expressions", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 172, 505, 185 ], "spans": [ { "bbox": [ 105, 172, 250, 185 ], "score": 1.0, "content": "and sign constraints, by generating", "type": "text" }, { "bbox": [ 250, 174, 262, 183 ], "score": 0.85, "content": "n _ { v }", "type": "inline_equation" }, { "bbox": [ 262, 172, 505, 185 ], "score": 1.0, "content": "variable names (letters) assumed strictly positive as well as", "type": "text" } ], "index": 5 }, { "bbox": [ 107, 182, 506, 198 ], "spans": [ { "bbox": [ 107, 185, 119, 194 ], "score": 0.83, "content": "n _ { n }", "type": "inline_equation" }, { "bbox": [ 119, 182, 295, 198 ], "score": 1.0, "content": "integers (for which we know the sign). For", "type": "text" }, { "bbox": [ 295, 183, 310, 194 ], "score": 0.87, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 310, 182, 445, 198 ], "score": 1.0, "content": "rounds, we compose elements of", "type": "text" }, { "bbox": [ 446, 184, 455, 193 ], "score": 0.81, "content": "E", "type": "inline_equation" }, { "bbox": [ 455, 182, 506, 198 ], "score": 1.0, "content": "using unary", "type": "text" } ], "index": 6 }, { "bbox": [ 107, 194, 505, 207 ], "spans": [ { "bbox": [ 107, 194, 208, 206 ], "score": 0.89, "content": "( l o g ( \\cdot ) , \\bar { l o g } ( 1 / \\cdot ) , s q r t ( \\cdot ) )", "type": "inline_equation" }, { "bbox": [ 208, 194, 293, 207 ], "score": 1.0, "content": "or binary operations", "type": "text" }, { "bbox": [ 294, 194, 401, 206 ], "score": 0.9, "content": "( + , - , \\times , / , \\wedge , m a x , m i n )", "type": "inline_equation" }, { "bbox": [ 401, 194, 505, 207 ], "score": 1.0, "content": "for which we can deduce", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 204, 505, 218 ], "spans": [ { "bbox": [ 105, 204, 505, 218 ], "score": 1.0, "content": "the sign based on the sign condition of the input expression(s) and re-inject the resulting expression", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 216, 500, 228 ], "spans": [ { "bbox": [ 105, 216, 194, 228 ], "score": 1.0, "content": "and sign constraint in", "type": "text" }, { "bbox": [ 195, 217, 204, 226 ], "score": 0.82, "content": "E", "type": "inline_equation" }, { "bbox": [ 204, 216, 287, 228 ], "score": 1.0, "content": ". This produces a set", "type": "text" }, { "bbox": [ 288, 217, 297, 226 ], "score": 0.83, "content": "E", "type": "inline_equation" }, { "bbox": [ 297, 216, 435, 228 ], "score": 1.0, "content": "of signed seed expressions of size", "type": "text" }, { "bbox": [ 435, 217, 496, 227 ], "score": 0.91, "content": "n _ { v } + n _ { n } + N _ { S }", "type": "inline_equation" }, { "bbox": [ 496, 216, 500, 228 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 9 } ], "index": 6 }, { "type": "text", "bbox": [ 106, 239, 505, 295 ], "lines": [ { "bbox": [ 106, 240, 505, 253 ], "spans": [ { "bbox": [ 106, 240, 505, 253 ], "score": 1.0, "content": "Inequality composition The second phase consists in generating inequalities from well known", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 249, 506, 265 ], "spans": [ { "bbox": [ 105, 249, 506, 265 ], "score": 1.0, "content": "inequality theorems (AM-GM, Trivial inequality, Cauchy-Schwarz, Bernoulli, Young, Hölder) taking", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 262, 506, 275 ], "spans": [ { "bbox": [ 105, 262, 278, 275 ], "score": 1.0, "content": "as input to these theorems expressions from", "type": "text" }, { "bbox": [ 278, 262, 287, 272 ], "score": 0.83, "content": "E", "type": "inline_equation" }, { "bbox": [ 288, 262, 506, 275 ], "score": 1.0, "content": "based on the sign constraints required for each theorem.", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 272, 505, 286 ], "spans": [ { "bbox": [ 106, 272, 261, 286 ], "score": 1.0, "content": "We finally compose these inequalities", "type": "text" }, { "bbox": [ 261, 273, 277, 284 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 278, 272, 505, 286 ], "score": 1.0, "content": "times using compositions theorems detailed in F.2. The", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 284, 506, 297 ], "spans": [ { "bbox": [ 105, 284, 319, 297 ], "score": 1.0, "content": "resulting inequality is a composed inequality of depth", "type": "text" }, { "bbox": [ 319, 284, 335, 295 ], "score": 0.9, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 335, 284, 374, 297 ], "score": 1.0, "content": "based on", "type": "text" }, { "bbox": [ 374, 284, 435, 295 ], "score": 0.92, "content": "n _ { v } + n _ { n } + N _ { S }", "type": "inline_equation" }, { "bbox": [ 435, 284, 506, 297 ], "score": 1.0, "content": "seed expressions.", "type": "text" } ], "index": 14 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 307, 504, 330 ], "lines": [ { "bbox": [ 105, 305, 504, 320 ], "spans": [ { "bbox": [ 105, 305, 504, 320 ], "score": 1.0, "content": "Simplification We finally post-process these inequalities so that they are parsable by Lean and run", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 318, 339, 331 ], "spans": [ { "bbox": [ 106, 318, 339, 331 ], "score": 1.0, "content": "them through Lean’s simp tactic for a final simplification.", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "text", "bbox": [ 107, 335, 505, 390 ], "lines": [ { "bbox": [ 106, 335, 505, 347 ], "spans": [ { "bbox": [ 106, 335, 123, 347 ], "score": 0.88, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 123, 335, 143, 347 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 143, 335, 158, 347 ], "score": 0.88, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 158, 335, 414, 347 ], "score": 1.0, "content": "together control for the difficulty of the resulting inequality.", "type": "text" }, { "bbox": [ 414, 335, 430, 346 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 430, 335, 505, 347 ], "score": 1.0, "content": "controls depth of", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 345, 505, 359 ], "spans": [ { "bbox": [ 105, 345, 183, 359 ], "score": 1.0, "content": "composition, while", "type": "text" }, { "bbox": [ 183, 347, 198, 357 ], "score": 0.87, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 199, 345, 505, 359 ], "score": 1.0, "content": "controls for obfuscation as it increases the complexity of the input expressions", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 357, 505, 370 ], "spans": [ { "bbox": [ 105, 357, 374, 370 ], "score": 1.0, "content": "to the composed inequalities. When sampling inequalities, we", "type": "text" }, { "bbox": [ 374, 358, 410, 368 ], "score": 0.91, "content": "n _ { n } ~ = ~ 4", "type": "inline_equation" }, { "bbox": [ 411, 357, 505, 370 ], "score": 1.0, "content": "and randomly sample", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 367, 506, 381 ], "spans": [ { "bbox": [ 106, 368, 159, 379 ], "score": 0.9, "content": "2 \\leq n _ { v } \\leq 8", "type": "inline_equation" }, { "bbox": [ 159, 367, 506, 381 ], "score": 1.0, "content": "at each generation. We report below examples of generated inequalities for various", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 377, 199, 392 ], "spans": [ { "bbox": [ 105, 377, 145, 392 ], "score": 1.0, "content": "values of", "type": "text" }, { "bbox": [ 145, 380, 161, 390 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 162, 377, 179, 392 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 180, 379, 194, 390 ], "score": 0.89, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 194, 377, 199, 392 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 21 } ], "index": 19 }, { "type": "title", "bbox": [ 107, 404, 333, 415 ], "lines": [ { "bbox": [ 105, 404, 334, 416 ], "spans": [ { "bbox": [ 105, 404, 334, 416 ], "score": 1.0, "content": "F.2 LIST OF INEQUALITY COMPOSITION THEOREMS", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 424, 505, 458 ], "lines": [ { "bbox": [ 105, 423, 505, 437 ], "spans": [ { "bbox": [ 105, 423, 505, 437 ], "score": 1.0, "content": "Below is the list of theorem names from mathlib that we use to compose inequalities together. One", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 434, 505, 449 ], "spans": [ { "bbox": [ 105, 434, 505, 449 ], "score": 1.0, "content": "third of the time, we only transform the current composed inequality with one of the following", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 446, 149, 459 ], "spans": [ { "bbox": [ 106, 446, 149, 459 ], "score": 1.0, "content": "theorems:", "type": "text" } ], "index": 25 } ], "index": 24 }, { "type": "text", "bbox": [ 133, 468, 244, 525 ], "lines": [ { "bbox": [ 133, 467, 195, 482 ], "spans": [ { "bbox": [ 133, 467, 195, 482 ], "score": 1.0, "content": "• neg_le_neg", "type": "text" } ], "index": 26 }, { "bbox": [ 133, 482, 194, 495 ], "spans": [ { "bbox": [ 133, 482, 194, 495 ], "score": 1.0, "content": "• inv_le_inv", "type": "text" } ], "index": 27 }, { "bbox": [ 134, 498, 244, 510 ], "spans": [ { "bbox": [ 134, 498, 244, 510 ], "score": 1.0, "content": "• mul_self_le_mul_self", "type": "text" } ], "index": 28 }, { "bbox": [ 133, 512, 225, 525 ], "spans": [ { "bbox": [ 133, 512, 225, 525 ], "score": 1.0, "content": "• div_le_one_of_le", "type": "text" } ], "index": 29 } ], "index": 27.5 }, { "type": "text", "bbox": [ 106, 535, 505, 558 ], "lines": [ { "bbox": [ 105, 533, 505, 549 ], "spans": [ { "bbox": [ 105, 533, 505, 549 ], "score": 1.0, "content": "We otherwise compose the current composed inequality with a newly generated inequality using the", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 545, 190, 560 ], "spans": [ { "bbox": [ 105, 545, 190, 560 ], "score": 1.0, "content": "following theorems:", "type": "text" } ], "index": 31 } ], "index": 30.5 }, { "type": "text", "bbox": [ 133, 568, 244, 640 ], "lines": [ { "bbox": [ 132, 567, 194, 579 ], "spans": [ { "bbox": [ 132, 567, 194, 579 ], "score": 1.0, "content": "• mul_le_mul", "type": "text" } ], "index": 32 }, { "bbox": [ 134, 583, 195, 594 ], "spans": [ { "bbox": [ 134, 583, 195, 594 ], "score": 1.0, "content": "• add_le_add", "type": "text" } ], "index": 33 }, { "bbox": [ 133, 597, 195, 610 ], "spans": [ { "bbox": [ 133, 597, 195, 610 ], "score": 1.0, "content": "• div_le_div", "type": "text" } ], "index": 34 }, { "bbox": [ 133, 612, 244, 625 ], "spans": [ { "bbox": [ 133, 612, 244, 625 ], "score": 1.0, "content": "• mul_le_mul_of_nonneg", "type": "text" } ], "index": 35 }, { "bbox": [ 133, 627, 219, 640 ], "spans": [ { "bbox": [ 133, 627, 219, 640 ], "score": 1.0, "content": "• le_mul_of_ratio", "type": "text" } ], "index": 36 } ], "index": 34 }, { "type": "text", "bbox": [ 106, 652, 179, 664 ], "lines": [ { "bbox": [ 105, 652, 180, 665 ], "spans": [ { "bbox": [ 105, 652, 180, 665 ], "score": 1.0, "content": "F.3 EXAMPLES", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "interline_equation", "bbox": [ 105, 672, 178, 706 ], "lines": [ { "bbox": [ 105, 672, 178, 706 ], "spans": [ { "bbox": [ 105, 672, 178, 706 ], "score": 0.59, "content": "\\begin{array} { c } { { N _ { D } = 0 \\ N _ { S } = 0 } } \\\\ { { { } } } \\\\ { { N _ { D } = 0 \\ N _ { S } = 4 } } \\end{array}", "type": "interline_equation", "image_path": "62ada2d1581d0d5d62e6363863a3b4c59bd9ef418ab9c0e3639ea5c4f7820f5a.jpg" } ] } ], "index": 38.5, "virtual_lines": [ { "bbox": [ 105, 672, 178, 689.0 ], "spans": [], "index": 38 }, { "bbox": [ 105, 689.0, 178, 706.0 ], "spans": [], "index": 39 } ] } ], "page_idx": 16, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 108, 81, 262, 94 ], "lines": [ { "bbox": [ 105, 80, 263, 96 ], 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"bbox": [ 105, 161, 506, 174 ], "spans": [ { "bbox": [ 105, 161, 351, 174 ], "score": 1.0, "content": "we track the sign. We start by initializing an expression set", "type": "text" }, { "bbox": [ 351, 162, 360, 171 ], "score": 0.8, "content": "E", "type": "inline_equation" }, { "bbox": [ 361, 161, 506, 174 ], "score": 1.0, "content": "composed of tuples of expressions", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 172, 505, 185 ], "spans": [ { "bbox": [ 105, 172, 250, 185 ], "score": 1.0, "content": "and sign constraints, by generating", "type": "text" }, { "bbox": [ 250, 174, 262, 183 ], "score": 0.85, "content": "n _ { v }", "type": "inline_equation" }, { "bbox": [ 262, 172, 505, 185 ], "score": 1.0, "content": "variable names (letters) assumed strictly positive as well as", "type": "text" } ], "index": 5 }, { "bbox": [ 107, 182, 506, 198 ], "spans": [ { "bbox": [ 107, 185, 119, 194 ], "score": 0.83, "content": "n _ { n }", "type": "inline_equation" }, { "bbox": [ 119, 182, 295, 198 ], "score": 1.0, "content": "integers (for which we know the sign). For", "type": "text" }, { "bbox": [ 295, 183, 310, 194 ], "score": 0.87, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 310, 182, 445, 198 ], "score": 1.0, "content": "rounds, we compose elements of", "type": "text" }, { "bbox": [ 446, 184, 455, 193 ], "score": 0.81, "content": "E", "type": "inline_equation" }, { "bbox": [ 455, 182, 506, 198 ], "score": 1.0, "content": "using unary", "type": "text" } ], "index": 6 }, { "bbox": [ 107, 194, 505, 207 ], "spans": [ { "bbox": [ 107, 194, 208, 206 ], "score": 0.89, "content": "( l o g ( \\cdot ) , \\bar { l o g } ( 1 / \\cdot ) , s q r t ( \\cdot ) )", "type": "inline_equation" }, { "bbox": [ 208, 194, 293, 207 ], "score": 1.0, "content": "or binary operations", "type": "text" }, { "bbox": [ 294, 194, 401, 206 ], "score": 0.9, "content": "( + , - , \\times , / , \\wedge , m a x , m i n )", "type": "inline_equation" }, { "bbox": [ 401, 194, 505, 207 ], "score": 1.0, "content": "for which we can deduce", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 204, 505, 218 ], "spans": [ { "bbox": [ 105, 204, 505, 218 ], "score": 1.0, "content": "the sign based on the sign condition of the input expression(s) and re-inject the resulting expression", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 216, 500, 228 ], "spans": [ { "bbox": [ 105, 216, 194, 228 ], "score": 1.0, "content": "and sign constraint in", "type": "text" }, { "bbox": [ 195, 217, 204, 226 ], "score": 0.82, "content": "E", "type": "inline_equation" }, { "bbox": [ 204, 216, 287, 228 ], "score": 1.0, "content": ". This produces a set", "type": "text" }, { "bbox": [ 288, 217, 297, 226 ], "score": 0.83, "content": "E", "type": "inline_equation" }, { "bbox": [ 297, 216, 435, 228 ], "score": 1.0, "content": "of signed seed expressions of size", "type": "text" }, { "bbox": [ 435, 217, 496, 227 ], "score": 0.91, "content": "n _ { v } + n _ { n } + N _ { S }", "type": "inline_equation" }, { "bbox": [ 496, 216, 500, 228 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 9 } ], "index": 6, "bbox_fs": [ 105, 151, 506, 228 ] }, { "type": "text", "bbox": [ 106, 239, 505, 295 ], "lines": [ { "bbox": [ 106, 240, 505, 253 ], "spans": [ { "bbox": [ 106, 240, 505, 253 ], "score": 1.0, "content": "Inequality composition The second phase consists in generating inequalities from well known", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 249, 506, 265 ], "spans": [ { "bbox": [ 105, 249, 506, 265 ], "score": 1.0, "content": "inequality theorems (AM-GM, Trivial inequality, Cauchy-Schwarz, Bernoulli, Young, Hölder) taking", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 262, 506, 275 ], "spans": [ { "bbox": [ 105, 262, 278, 275 ], "score": 1.0, "content": "as input to these theorems expressions from", "type": "text" }, { "bbox": [ 278, 262, 287, 272 ], "score": 0.83, "content": "E", "type": "inline_equation" }, { "bbox": [ 288, 262, 506, 275 ], "score": 1.0, "content": "based on the sign constraints required for each theorem.", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 272, 505, 286 ], "spans": [ { "bbox": [ 106, 272, 261, 286 ], "score": 1.0, "content": "We finally compose these inequalities", "type": "text" }, { "bbox": [ 261, 273, 277, 284 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 278, 272, 505, 286 ], "score": 1.0, "content": "times using compositions theorems detailed in F.2. The", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 284, 506, 297 ], "spans": [ { "bbox": [ 105, 284, 319, 297 ], "score": 1.0, "content": "resulting inequality is a composed inequality of depth", "type": "text" }, { "bbox": [ 319, 284, 335, 295 ], "score": 0.9, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 335, 284, 374, 297 ], "score": 1.0, "content": "based on", "type": "text" }, { "bbox": [ 374, 284, 435, 295 ], "score": 0.92, "content": "n _ { v } + n _ { n } + N _ { S }", "type": "inline_equation" }, { "bbox": [ 435, 284, 506, 297 ], "score": 1.0, "content": "seed expressions.", "type": "text" } ], "index": 14 } ], "index": 12, "bbox_fs": [ 105, 240, 506, 297 ] }, { "type": "text", "bbox": [ 107, 307, 504, 330 ], "lines": [ { "bbox": [ 105, 305, 504, 320 ], "spans": [ { "bbox": [ 105, 305, 504, 320 ], "score": 1.0, "content": "Simplification We finally post-process these inequalities so that they are parsable by Lean and run", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 318, 339, 331 ], "spans": [ { "bbox": [ 106, 318, 339, 331 ], "score": 1.0, "content": "them through Lean’s simp tactic for a final simplification.", "type": "text" } ], "index": 16 } ], "index": 15.5, "bbox_fs": [ 105, 305, 504, 331 ] }, { "type": "text", "bbox": [ 107, 335, 505, 390 ], "lines": [ { "bbox": [ 106, 335, 505, 347 ], "spans": [ { "bbox": [ 106, 335, 123, 347 ], "score": 0.88, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 123, 335, 143, 347 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 143, 335, 158, 347 ], "score": 0.88, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 158, 335, 414, 347 ], "score": 1.0, "content": "together control for the difficulty of the resulting inequality.", "type": "text" }, { "bbox": [ 414, 335, 430, 346 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 430, 335, 505, 347 ], "score": 1.0, "content": "controls depth of", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 345, 505, 359 ], "spans": [ { "bbox": [ 105, 345, 183, 359 ], "score": 1.0, "content": "composition, while", "type": "text" }, { "bbox": [ 183, 347, 198, 357 ], "score": 0.87, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 199, 345, 505, 359 ], "score": 1.0, "content": "controls for obfuscation as it increases the complexity of the input expressions", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 357, 505, 370 ], "spans": [ { "bbox": [ 105, 357, 374, 370 ], "score": 1.0, "content": "to the composed inequalities. When sampling inequalities, we", "type": "text" }, { "bbox": [ 374, 358, 410, 368 ], "score": 0.91, "content": "n _ { n } ~ = ~ 4", "type": "inline_equation" }, { "bbox": [ 411, 357, 505, 370 ], "score": 1.0, "content": "and randomly sample", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 367, 506, 381 ], "spans": [ { "bbox": [ 106, 368, 159, 379 ], "score": 0.9, "content": "2 \\leq n _ { v } \\leq 8", "type": "inline_equation" }, { "bbox": [ 159, 367, 506, 381 ], "score": 1.0, "content": "at each generation. We report below examples of generated inequalities for various", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 377, 199, 392 ], "spans": [ { "bbox": [ 105, 377, 145, 392 ], "score": 1.0, "content": "values of", "type": "text" }, { "bbox": [ 145, 380, 161, 390 ], "score": 0.89, "content": "N _ { D }", "type": "inline_equation" }, { "bbox": [ 162, 377, 179, 392 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 180, 379, 194, 390 ], "score": 0.89, "content": "N _ { S }", "type": "inline_equation" }, { "bbox": [ 194, 377, 199, 392 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 21 } ], "index": 19, "bbox_fs": [ 105, 335, 506, 392 ] }, { "type": "title", "bbox": [ 107, 404, 333, 415 ], "lines": [ { "bbox": [ 105, 404, 334, 416 ], "spans": [ { "bbox": [ 105, 404, 334, 416 ], "score": 1.0, "content": "F.2 LIST OF INEQUALITY COMPOSITION THEOREMS", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 424, 505, 458 ], "lines": [ { "bbox": [ 105, 423, 505, 437 ], "spans": [ { "bbox": [ 105, 423, 505, 437 ], "score": 1.0, "content": "Below is the list of theorem names from mathlib that we use to compose inequalities together. One", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 434, 505, 449 ], "spans": [ { "bbox": [ 105, 434, 505, 449 ], "score": 1.0, "content": "third of the time, we only transform the current composed inequality with one of the following", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 446, 149, 459 ], "spans": [ { "bbox": [ 106, 446, 149, 459 ], "score": 1.0, "content": "theorems:", "type": "text" } ], "index": 25 } ], "index": 24, "bbox_fs": [ 105, 423, 505, 459 ] }, { "type": "list", "bbox": [ 133, 468, 244, 525 ], "lines": [ { "bbox": [ 133, 467, 195, 482 ], "spans": [ { "bbox": [ 133, 467, 195, 482 ], "score": 1.0, "content": "• neg_le_neg", "type": "text" } ], "index": 26, "is_list_start_line": true }, { "bbox": [ 133, 482, 194, 495 ], "spans": [ { "bbox": [ 133, 482, 194, 495 ], "score": 1.0, "content": "• inv_le_inv", "type": "text" } ], "index": 27, "is_list_start_line": true }, { "bbox": [ 134, 498, 244, 510 ], "spans": [ { "bbox": [ 134, 498, 244, 510 ], "score": 1.0, "content": "• mul_self_le_mul_self", "type": "text" } ], "index": 28, "is_list_start_line": true }, { "bbox": [ 133, 512, 225, 525 ], "spans": [ { "bbox": [ 133, 512, 225, 525 ], "score": 1.0, "content": "• div_le_one_of_le", "type": "text" } ], "index": 29, "is_list_start_line": true } ], "index": 27.5, "bbox_fs": [ 133, 467, 244, 525 ] }, { "type": "text", "bbox": [ 106, 535, 505, 558 ], "lines": [ { "bbox": [ 105, 533, 505, 549 ], "spans": [ { "bbox": [ 105, 533, 505, 549 ], "score": 1.0, "content": "We otherwise compose the current composed inequality with a newly generated inequality using the", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 545, 190, 560 ], "spans": [ { "bbox": [ 105, 545, 190, 560 ], "score": 1.0, "content": "following theorems:", "type": "text" } ], "index": 31 } ], "index": 30.5, "bbox_fs": [ 105, 533, 505, 560 ] }, { "type": "list", "bbox": [ 133, 568, 244, 640 ], "lines": [ { "bbox": [ 132, 567, 194, 579 ], "spans": [ { "bbox": [ 132, 567, 194, 579 ], "score": 1.0, "content": "• mul_le_mul", "type": "text" } ], "index": 32, "is_list_start_line": true }, { "bbox": [ 134, 583, 195, 594 ], "spans": [ { "bbox": [ 134, 583, 195, 594 ], "score": 1.0, "content": "• add_le_add", "type": "text" } ], "index": 33, "is_list_start_line": true }, { "bbox": [ 133, 597, 195, 610 ], "spans": [ { "bbox": [ 133, 597, 195, 610 ], "score": 1.0, "content": "• div_le_div", "type": "text" } ], "index": 34, "is_list_start_line": true }, { "bbox": [ 133, 612, 244, 625 ], "spans": [ { "bbox": [ 133, 612, 244, 625 ], "score": 1.0, "content": "• mul_le_mul_of_nonneg", "type": "text" } ], "index": 35, "is_list_start_line": true }, { "bbox": [ 133, 627, 219, 640 ], "spans": [ { "bbox": [ 133, 627, 219, 640 ], "score": 1.0, "content": "• le_mul_of_ratio", "type": "text" } ], "index": 36, "is_list_start_line": true } ], "index": 34, "bbox_fs": [ 132, 567, 244, 640 ] }, { "type": "text", "bbox": [ 106, 652, 179, 664 ], "lines": [ { "bbox": [ 105, 652, 180, 665 ], "spans": [ { "bbox": [ 105, 652, 180, 665 ], "score": 1.0, "content": "F.3 EXAMPLES", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 105, 652, 180, 665 ] }, { "type": "interline_equation", "bbox": [ 105, 672, 178, 706 ], "lines": [ { "bbox": [ 105, 672, 178, 706 ], "spans": [ { "bbox": [ 105, 672, 178, 706 ], "score": 0.59, "content": "\\begin{array} { c } { { N _ { D } = 0 \\ N _ { S } = 0 } } \\\\ { { { } } } \\\\ { { N _ { D } = 0 \\ N _ { S } = 4 } } \\end{array}", "type": "interline_equation", "image_path": "62ada2d1581d0d5d62e6363863a3b4c59bd9ef418ab9c0e3639ea5c4f7820f5a.jpg" } ] } ], "index": 38.5, "virtual_lines": [ { "bbox": [ 105, 672, 178, 689.0 ], "spans": [], "index": 38 }, { "bbox": [ 105, 689.0, 178, 706.0 ], "spans": [], "index": 39 } ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 106, 80, 502, 192 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 80, 502, 192 ], "group_id": 0, "lines": [ { "bbox": [ 106, 80, 502, 192 ], "spans": [ { "bbox": [ 106, 80, 502, 192 ], "score": 0.936, "html": "
CompositionsAmGm a b (67:R)((1:R)/(10:R)）((1:R)/(10:R)) ((8:R)/(10:R))
Statementtheorem synthetic_ineq_nb_seed_var_0_depth_0_p_1 (ab:R) (h0:0<a) (h1:0<b): (67:R)^（(8:R)/(10:R)）*b^(10:R)-1 * a^(10:R)-1≤ (8:R)/(10:R)* (67:R) + (10:R)-1 * a+b* (10:R)-1 := sorry
", "type": "table", "image_path": "9e10b1e70d0e72693181cb238f1b464e1e68fe20e052cdfa2e99b204d2901854.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 106, 80, 502, 117.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 106, 117.33333333333334, 502, 154.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 106, 154.66666666666669, 502, 192.00000000000003 ], "spans": [], "index": 2 } ] } ], "index": 1 }, { "type": "table", "bbox": [ 107, 203, 502, 295 ], "blocks": [ { "type": "table_body", "bbox": [ 107, 203, 502, 295 ], "group_id": 1, "lines": [ { "bbox": [ 107, 203, 502, 295 ], "spans": [ { "bbox": [ 107, 203, 502, 295 ], "score": 0.951, "html": "
CompositionsSqnonneg a ((a)+((-68:R)))
Statementtheorem synthetic_ineq_nb_seed_var_4_depth_0_p_4 (ab:R) (h0 :0<a) (h1:0<b): (2:R)*(a*（a+-(68:R))）≤ (a +-(68:R))^2 +a^2 := sorry
", "type": "table", "image_path": "cda1dd47de04b3195de0b885534a3a8b3e89f6ee593c1fb752cd8565b1b4fbcc.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 107, 203, 502, 233.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 107, 233.66666666666666, 502, 264.3333333333333 ], "spans": [], "index": 4 }, { "bbox": [ 107, 264.3333333333333, 502, 295.0 ], "spans": [], "index": 5 } ] } ], "index": 4 }, { "type": "interline_equation", "bbox": [ 106, 315, 177, 328 ], "lines": [ { "bbox": [ 106, 315, 177, 328 ], "spans": [ { "bbox": [ 106, 315, 177, 328 ], "score": 0.76, "content": "N _ { D } = 4 N _ { S } = 4", "type": "interline_equation", "image_path": "62333298a80a5348f26cb368481a17f0226de7704b32baace54b636652151ff4.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 106, 315, 177, 328 ], "spans": [], "index": 6 } ] }, { "type": "table", "bbox": [ 106, 341, 502, 612 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 341, 502, 612 ], "group_id": 2, "lines": [ { "bbox": [ 106, 341, 502, 612 ], "spans": [ { "bbox": [ 106, 341, 502, 612 ], "score": 0.982, "html": "
CompositionsAddLeAdd Bernoulli 99 c AddLeAdd SelfDivConst（(a）/（f)）6 LeMulOfRatio SelfDivConst c 70 DivLeDiv Cauchy（(a）/(f)）dc(log（((59:R）+ f))) Young((a)/(f)）a((3:R)/(2:R)）((3:R)/(1:R)) theorem synthetic_ineq_nb_seed_var_4_depth_4_p_13
Statement(abcdef:R) (h0 :0<a) (h1 :0<b) (h2 :0<c) (h3:0<d) (h4 :0<e) (h5:0<f): (1:R)+(99:R)*c+(a/f/(6:R)+a*(a/f)/ （(d^2+a^2/f^2)* (real.log（(59:R）+f)^2+c^2))）≤ （(a/f）^((3:R)/(2:R)）/（(3:R)/(2:R)）+ a^3/(3:R)）/ (real.log（(59:R）+f）*d+a/ f*c)^2 * (c/(c/(70:R))）+a/f+(c+(1:R)）^99 := sorry
", "type": "table", "image_path": "359bcaa63a105aba94a8629d110a5ab20fb19c87dcd372ce203ed76c8a8d1c57.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 106, 341, 502, 431.3333333333333 ], "spans": [], "index": 7 }, { "bbox": [ 106, 431.3333333333333, 502, 521.6666666666666 ], "spans": [], "index": 8 }, { "bbox": [ 106, 521.6666666666666, 502, 612.0 ], "spans": [], "index": 9 } ] } ], "index": 8 } ], "page_idx": 17, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 763 ], "spans": [ { "bbox": [ 299, 750, 312, 763 ], "score": 1.0, "content": "18", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 106, 80, 502, 192 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 80, 502, 192 ], "group_id": 0, "lines": [ { "bbox": [ 106, 80, 502, 192 ], "spans": [ { "bbox": [ 106, 80, 502, 192 ], "score": 0.936, "html": "
CompositionsAmGm a b (67:R)((1:R)/(10:R)）((1:R)/(10:R)) ((8:R)/(10:R))
Statementtheorem synthetic_ineq_nb_seed_var_0_depth_0_p_1 (ab:R) (h0:0<a) (h1:0<b): (67:R)^（(8:R)/(10:R)）*b^(10:R)-1 * a^(10:R)-1≤ (8:R)/(10:R)* (67:R) + (10:R)-1 * a+b* (10:R)-1 := sorry
", "type": "table", "image_path": "9e10b1e70d0e72693181cb238f1b464e1e68fe20e052cdfa2e99b204d2901854.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 106, 80, 502, 117.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 106, 117.33333333333334, 502, 154.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 106, 154.66666666666669, 502, 192.00000000000003 ], "spans": [], "index": 2 } ] } ], "index": 1 }, { "type": "table", "bbox": [ 107, 203, 502, 295 ], "blocks": [ { "type": "table_body", "bbox": [ 107, 203, 502, 295 ], "group_id": 1, "lines": [ { "bbox": [ 107, 203, 502, 295 ], "spans": [ { "bbox": [ 107, 203, 502, 295 ], "score": 0.951, "html": "
CompositionsSqnonneg a ((a)+((-68:R)))
Statementtheorem synthetic_ineq_nb_seed_var_4_depth_0_p_4 (ab:R) (h0 :0<a) (h1:0<b): (2:R)*(a*（a+-(68:R))）≤ (a +-(68:R))^2 +a^2 := sorry
", "type": "table", "image_path": "cda1dd47de04b3195de0b885534a3a8b3e89f6ee593c1fb752cd8565b1b4fbcc.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 107, 203, 502, 233.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 107, 233.66666666666666, 502, 264.3333333333333 ], "spans": [], "index": 4 }, { "bbox": [ 107, 264.3333333333333, 502, 295.0 ], "spans": [], "index": 5 } ] } ], "index": 4 }, { "type": "interline_equation", "bbox": [ 106, 315, 177, 328 ], "lines": [ { "bbox": [ 106, 315, 177, 328 ], "spans": [ { "bbox": [ 106, 315, 177, 328 ], "score": 0.76, "content": "N _ { D } = 4 N _ { S } = 4", "type": "interline_equation", "image_path": "62333298a80a5348f26cb368481a17f0226de7704b32baace54b636652151ff4.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 106, 315, 177, 328 ], "spans": [], "index": 6 } ] }, { "type": "table", "bbox": [ 106, 341, 502, 612 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 341, 502, 612 ], "group_id": 2, "lines": [ { "bbox": [ 106, 341, 502, 612 ], "spans": [ { "bbox": [ 106, 341, 502, 612 ], "score": 0.982, "html": "
CompositionsAddLeAdd Bernoulli 99 c AddLeAdd SelfDivConst（(a）/（f)）6 LeMulOfRatio SelfDivConst c 70 DivLeDiv Cauchy（(a）/(f)）dc(log（((59:R）+ f))) Young((a)/(f)）a((3:R)/(2:R)）((3:R)/(1:R)) theorem synthetic_ineq_nb_seed_var_4_depth_4_p_13
Statement(abcdef:R) (h0 :0<a) (h1 :0<b) (h2 :0<c) (h3:0<d) (h4 :0<e) (h5:0<f): (1:R)+(99:R)*c+(a/f/(6:R)+a*(a/f)/ （(d^2+a^2/f^2)* (real.log（(59:R）+f)^2+c^2))）≤ （(a/f）^((3:R)/(2:R)）/（(3:R)/(2:R)）+ a^3/(3:R)）/ (real.log（(59:R）+f）*d+a/ f*c)^2 * (c/(c/(70:R))）+a/f+(c+(1:R)）^99 := sorry
", "type": "table", "image_path": "359bcaa63a105aba94a8629d110a5ab20fb19c87dcd372ce203ed76c8a8d1c57.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 106, 341, 502, 431.3333333333333 ], "spans": [], "index": 7 }, { "bbox": [ 106, 431.3333333333333, 502, 521.6666666666666 ], "spans": [], "index": 8 }, { "bbox": [ 106, 521.6666666666666, 502, 612.0 ], "spans": [], "index": 9 } ] } ], "index": 8 } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 81, 248, 94 ], "lines": [ { "bbox": [ 105, 79, 250, 97 ], "spans": [ { "bbox": [ 105, 79, 250, 97 ], "score": 1.0, "content": "G MINIF2F-CURRICULUM", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 105, 409, 118 ], "lines": [ { "bbox": [ 105, 104, 411, 120 ], "spans": [ { "bbox": [ 105, 104, 411, 120 ], "score": 1.0, "content": "The 327 statements of miniF2F-curriculum3 are manually formalized from:", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 133, 128, 505, 182 ], "lines": [ { "bbox": [ 133, 128, 505, 141 ], "spans": [ { "bbox": [ 133, 128, 505, 141 ], "score": 1.0, "content": "• AOPS Books (Lehoczky & Rusczyk, a;b): 302 examples and exercises. 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Statementlemma comap_eq_of_inverse {f：filter α} {g：filter β}{Φ:α→β} (φ:β→α)(eq:γ○Φ=id)(hΦ : tendsto f g)(hψ :tendsto 𝜑 g f):comap Φ g = f :=
Ground-truthbeginrefine((comap_mono $map_le_iff_le_comap.1 h).trans_).antisymm(map_le_iff_le_comap.1 hΦ),rw [comap_comap，eq，comap_id],exact le_rflend
Model proofbeginrefine le_antisymm _ (filter.map_le_iff_le_comap.1 h),refine 入s hs,-,rw mem_comap,use [ -1' s,h hs],rw[← preimage_comp，eq，preimage_id]end
", "type": "table", "image_path": "bb7b06cc92c4d142981f85598ecd5c22f4c64babbec1d9c3a63d2e7f164d555d.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 106, 167, 502, 238.0 ], "spans": [], "index": 4 }, { "bbox": [ 106, 238.0, 502, 309.0 ], "spans": [], "index": 5 }, { "bbox": [ 106, 309.0, 502, 380.0 ], "spans": [], "index": 6 } ] } ], "index": 5 } ], "page_idx": 19, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 294, 38 ], "spans": [ { "bbox": [ 106, 26, 294, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 749, 313, 764 ], "spans": [ { "bbox": [ 298, 749, 313, 764 ], "score": 1.0, "content": "", "type": "text", "height": 15, "width": 15 } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 106, 81, 317, 94 ], "lines": [ { "bbox": [ 105, 80, 317, 95 ], "spans": [ { "bbox": [ 105, 80, 317, 95 ], "score": 1.0, "content": "I EXAMPLE PROOFS FROM mathlib-train", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 106, 504, 128 ], "lines": [ { "bbox": [ 105, 105, 505, 119 ], "spans": [ { "bbox": [ 105, 105, 505, 119 ], "score": 1.0, "content": "We present in this section original proofs found by our models from mathlib-train, compared with", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 116, 214, 130 ], "spans": [ { "bbox": [ 105, 116, 214, 130 ], "score": 1.0, "content": "their ground-truth version.", "type": "text" } ], "index": 2 } ], "index": 1.5, "bbox_fs": [ 105, 105, 505, 130 ] }, { "type": "text", "bbox": [ 107, 142, 203, 154 ], "lines": [ { "bbox": [ 106, 142, 204, 155 ], "spans": [ { "bbox": [ 106, 142, 204, 155 ], "score": 1.0, "content": "comap_eq_of_inverse", "type": "text" } ], "index": 3 } ], "index": 3, "bbox_fs": [ 106, 142, 204, 155 ] }, { "type": "table", "bbox": [ 106, 167, 502, 380 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 167, 502, 380 ], "group_id": 0, "lines": [ { "bbox": [ 106, 167, 502, 380 ], "spans": [ { "bbox": [ 106, 167, 502, 380 ], "score": 0.983, "html": "
Statementlemma comap_eq_of_inverse {f：filter α} {g：filter β}{Φ:α→β} (φ:β→α)(eq:γ○Φ=id)(hΦ : tendsto f g)(hψ :tendsto 𝜑 g f):comap Φ g = f :=
Ground-truthbeginrefine((comap_mono $map_le_iff_le_comap.1 h).trans_).antisymm(map_le_iff_le_comap.1 hΦ),rw [comap_comap，eq，comap_id],exact le_rflend
Model proofbeginrefine le_antisymm _ (filter.map_le_iff_le_comap.1 h),refine 入s hs,-,rw mem_comap,use [ -1' s,h hs],rw[← preimage_comp，eq，preimage_id]end
", "type": "table", "image_path": "bb7b06cc92c4d142981f85598ecd5c22f4c64babbec1d9c3a63d2e7f164d555d.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 106, 167, 502, 238.0 ], "spans": [], "index": 4 }, { "bbox": [ 106, 238.0, 502, 309.0 ], "spans": [], "index": 5 }, { "bbox": [ 106, 309.0, 502, 380.0 ], "spans": [], "index": 6 } ] } ], "index": 5 } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 84, 223, 94 ], "lines": [ { "bbox": [ 106, 83, 223, 96 ], "spans": [ { "bbox": [ 106, 83, 223, 96 ], "score": 1.0, "content": "sum_range_sub_sum_range", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table", "bbox": [ 106, 108, 502, 370 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 108, 502, 370 ], "group_id": 0, "lines": [ { "bbox": [ 106, 108, 502, 370 ], "spans": [ { "bbox": [ 106, 108, 502, 370 ], "score": 0.979, "html": "
Statementlemma sum_range_sub_sum_range {α :Type*}[add_comm_group α]{f：N→α}{nm：N}(hnm:n≤m）:∑k in range m，fk-∑kin range n，fk =∑k in (range m).filter (λk，n≤k)，f k :=
Ground-truthbeginrw[← sum_sdiff(@filter_subset _（λ k，n ≤k）_(range m))，sub_eq_iff_eq_add,← eq_sub_iff_add_eq，add_sub_cancel'],refine finset.sum_congr(finset.ext $入 a，< h，by simp at *；finish,入 h，have ham ：a<m :=lt_of_lt_of_le (mem_range.1 h）hnm,by simp * at *>)(λ__，rfl)end
Model proofbeginrw[← sum_Ico_eq_sub_ hnm],congr,apply finset.ext,simp [Ico.mem，*],tautoend
", "type": "table", "image_path": "4ee9d80dac27360093ab4f885aa6f4e5a7271a0b3760144ab536a230c1c83e0e.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 108, 502, 195.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 195.33333333333331, 502, 282.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 282.66666666666663, 502, 369.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 2 } ], "page_idx": 20, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 764 ], "spans": [ { "bbox": [ 298, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 14 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 84, 223, 94 ], "lines": [ { "bbox": [ 106, 83, 223, 96 ], "spans": [ { "bbox": [ 106, 83, 223, 96 ], "score": 1.0, "content": "sum_range_sub_sum_range", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 106, 83, 223, 96 ] }, { "type": "table", "bbox": [ 106, 108, 502, 370 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 108, 502, 370 ], "group_id": 0, "lines": [ { "bbox": [ 106, 108, 502, 370 ], "spans": [ { "bbox": [ 106, 108, 502, 370 ], "score": 0.979, "html": "
Statementlemma sum_range_sub_sum_range {α :Type*}[add_comm_group α]{f：N→α}{nm：N}(hnm:n≤m）:∑k in range m，fk-∑kin range n，fk =∑k in (range m).filter (λk，n≤k)，f k :=
Ground-truthbeginrw[← sum_sdiff(@filter_subset _（λ k，n ≤k）_(range m))，sub_eq_iff_eq_add,← eq_sub_iff_add_eq，add_sub_cancel'],refine finset.sum_congr(finset.ext $入 a，< h，by simp at *；finish,入 h，have ham ：a<m :=lt_of_lt_of_le (mem_range.1 h）hnm,by simp * at *>)(λ__，rfl)end
Model proofbeginrw[← sum_Ico_eq_sub_ hnm],congr,apply finset.ext,simp [Ico.mem，*],tautoend
", "type": "table", "image_path": "4ee9d80dac27360093ab4f885aa6f4e5a7271a0b3760144ab536a230c1c83e0e.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 108, 502, 195.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 195.33333333333331, 502, 282.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 282.66666666666663, 502, 369.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 2 } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 83, 188, 94 ], "lines": [ { "bbox": [ 105, 82, 189, 95 ], "spans": [ { "bbox": [ 105, 82, 189, 95 ], "score": 1.0, "content": "prod_inv_distrib", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table", "bbox": [ 106, 107, 502, 360 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 107, 502, 360 ], "group_id": 0, "lines": [ { "bbox": [ 106, 107, 502, 360 ], "spans": [ { "bbox": [ 106, 107, 502, 360 ], "score": 0.981, "html": "
Statementlemma prod_inv_distrib :(II× in s，(f x)-1)=①I×in s,f x)-1 :=
Ground-truthbeginclassical,by_casesh:×∈s，fx=0,{simpa [prod_eq_zero_iff.mpr h，prod_eq_zero_iff]using h },{ push_neg at h,have h' := prod_ne_zero_iff.mpr h,have hf :∀x ∈ s，(f x)-1 * fx=1 :=λx hx,inv_mul_cancel (h × hx)，apply mul_right_cancel'h',simp[h，h'，← finset.prod_mul_distrib,prod_congr rfl hf]}end
Model proofbeginclassical；induction s usingfinset.induction_on with a s has ih,{simp，},simp only[has，prod_insert has，mul_inv_rev'],finishend
", "type": "table", "image_path": "16a85463891c3417c605eca6f5bcfb2d7e10f77e46a2187e1d26a9629e67affa.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 107, 502, 191.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 191.33333333333331, 502, 275.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 275.66666666666663, 502, 359.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 2 } ], "page_idx": 21, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "22", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 83, 188, 94 ], "lines": [ { "bbox": [ 105, 82, 189, 95 ], "spans": [ { "bbox": [ 105, 82, 189, 95 ], "score": 1.0, "content": "prod_inv_distrib", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 105, 82, 189, 95 ] }, { "type": "table", "bbox": [ 106, 107, 502, 360 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 107, 502, 360 ], "group_id": 0, "lines": [ { "bbox": [ 106, 107, 502, 360 ], "spans": [ { "bbox": [ 106, 107, 502, 360 ], "score": 0.981, "html": "
Statementlemma prod_inv_distrib :(II× in s，(f x)-1)=①I×in s,f x)-1 :=
Ground-truthbeginclassical,by_casesh:×∈s，fx=0,{simpa [prod_eq_zero_iff.mpr h，prod_eq_zero_iff]using h },{ push_neg at h,have h' := prod_ne_zero_iff.mpr h,have hf :∀x ∈ s，(f x)-1 * fx=1 :=λx hx,inv_mul_cancel (h × hx)，apply mul_right_cancel'h',simp[h，h'，← finset.prod_mul_distrib,prod_congr rfl hf]}end
Model proofbeginclassical；induction s usingfinset.induction_on with a s has ih,{simp，},simp only[has，prod_insert has，mul_inv_rev'],finishend
", "type": "table", "image_path": "16a85463891c3417c605eca6f5bcfb2d7e10f77e46a2187e1d26a9629e67affa.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 107, 502, 191.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 191.33333333333331, 502, 275.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 275.66666666666663, 502, 359.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 2 } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 106, 81, 414, 94 ], "lines": [ { "bbox": [ 105, 81, 415, 96 ], "spans": [ { "bbox": [ 105, 81, 415, 96 ], "score": 1.0, "content": "J EXAMPLE PROOFS FROM miniF2F-{test, valid, curriculum}", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 106, 504, 128 ], "lines": [ { "bbox": [ 105, 105, 506, 119 ], "spans": [ { "bbox": [ 105, 105, 506, 119 ], "score": 1.0, "content": "We present in this section proofs found by our models from miniF2F-{test, valid, curriculum},", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 117, 419, 130 ], "spans": [ { "bbox": [ 106, 117, 419, 130 ], "score": 1.0, "content": "demonstrating some of the capabilities emerging from our training procedure.", "type": "text" } ], "index": 2 } ], "index": 1.5 }, { "type": "title", "bbox": [ 108, 142, 282, 153 ], "lines": [ { "bbox": [ 105, 141, 285, 155 ], "spans": [ { "bbox": [ 105, 141, 285, 155 ], "score": 1.0, "content": "J.1 QUALITATIVE ANALYSIS OF PROOFS", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "text", "bbox": [ 107, 163, 505, 207 ], "lines": [ { "bbox": [ 106, 163, 505, 175 ], "spans": [ { "bbox": [ 106, 163, 505, 175 ], "score": 1.0, "content": "We provide qualitative insights in the nature of the proofs found by our models, which we believe", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 173, 506, 186 ], "spans": [ { "bbox": [ 105, 173, 506, 186 ], "score": 1.0, "content": "are useful to build a better intuition of their capabilities beyond pass rate numbers. Throughout this", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 185, 506, 197 ], "spans": [ { "bbox": [ 105, 185, 506, 197 ], "score": 1.0, "content": "section, we refer to statements and solutions found by our models that are presented in Appendix J", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 196, 356, 209 ], "spans": [ { "bbox": [ 105, 196, 356, 209 ], "score": 1.0, "content": "along with comments describing the specificity of each proof.", "type": "text" } ], "index": 7 } ], "index": 5.5 }, { "type": "text", "bbox": [ 107, 212, 505, 257 ], "lines": [ { "bbox": [ 105, 212, 505, 225 ], "spans": [ { "bbox": [ 105, 212, 505, 225 ], "score": 1.0, "content": "First, we observe that a large number of olympiad problems that are designed to be computationally", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 224, 505, 236 ], "spans": [ { "bbox": [ 106, 224, 505, 236 ], "score": 1.0, "content": "challenging for humans are rendered trivial for our models through the use of Lean tactics. As an", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 234, 507, 248 ], "spans": [ { "bbox": [ 105, 234, 507, 248 ], "score": 1.0, "content": "example, mathd_numbertheory_447 which is not necessarily considered straightforward for humans,", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 246, 384, 257 ], "spans": [ { "bbox": [ 105, 246, 384, 257 ], "score": 1.0, "content": "can be closed in Lean by a simple refl (proof found by our models).", "type": "text" } ], "index": 11 } ], "index": 9.5 }, { "type": "text", "bbox": [ 107, 262, 505, 361 ], "lines": [ { "bbox": [ 106, 262, 505, 274 ], "spans": [ { "bbox": [ 106, 262, 505, 274 ], "score": 1.0, "content": "In recent years, Lean’s mathlib community has developed high-powered tactics such as", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 273, 506, 286 ], "spans": [ { "bbox": [ 106, 273, 506, 286 ], "score": 1.0, "content": "linarith/nlinarith (solves (non)linear inequalities), norm_num (normalizes numerical expres-", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 285, 505, 297 ], "spans": [ { "bbox": [ 106, 285, 505, 297 ], "score": 1.0, "content": "sions), simp (simplifies goals and hypotheses) and ring (normalizes expressions in a ring). These", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 295, 504, 307 ], "spans": [ { "bbox": [ 106, 295, 504, 307 ], "score": 1.0, "content": "tactics can be used with arguments to guide their underlying search procedure. As mentioned in", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 306, 506, 319 ], "spans": [ { "bbox": [ 105, 306, 506, 319 ], "score": 1.0, "content": "Zheng et al. (2022), we confirm here that our models acquire advanced capabilities to leverage these", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 317, 505, 330 ], "spans": [ { "bbox": [ 105, 317, 505, 330 ], "score": 1.0, "content": "high-level tactics by providing exogenous arguments which are not present in the current tactic", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 327, 506, 342 ], "spans": [ { "bbox": [ 105, 327, 506, 342 ], "score": 1.0, "content": "state. The generation of these exogenous arguments through language modeling seems to require a", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 338, 505, 352 ], "spans": [ { "bbox": [ 105, 338, 505, 352 ], "score": 1.0, "content": "non-trivial amount of mathematical intuition. imo_1964_p2, imo_1961_p1 and aime_1990_p15 are", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 350, 223, 363 ], "spans": [ { "bbox": [ 105, 350, 223, 363 ], "score": 1.0, "content": "good examples of such uses.", "type": "text" } ], "index": 20 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 367, 505, 455 ], "lines": [ { "bbox": [ 105, 366, 505, 379 ], "spans": [ { "bbox": [ 105, 366, 505, 379 ], "score": 1.0, "content": "We have also observed a number of proofs that require multiple non-trivial reasoning steps through the", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 378, 506, 390 ], "spans": [ { "bbox": [ 105, 378, 506, 390 ], "score": 1.0, "content": "use of lower-level tactics such as use, have, or by_cases that generally involve producing a witness or", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 388, 506, 402 ], "spans": [ { "bbox": [ 105, 388, 506, 402 ], "score": 1.0, "content": "chaining implications, requiring the generation of context specific exogenous terms. These interesting", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 399, 505, 413 ], "spans": [ { "bbox": [ 105, 399, 505, 413 ], "score": 1.0, "content": "reasoning steps are structurally different from simple normalization, simplification and rewriting", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 411, 505, 423 ], "spans": [ { "bbox": [ 105, 411, 505, 423 ], "score": 1.0, "content": "of hypotheses or goals because they heavily rely on our models ability to generate meaningful", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 421, 506, 434 ], "spans": [ { "bbox": [ 105, 421, 506, 434 ], "score": 1.0, "content": "cuts or witnesses. This capability is, in our opinion, the most exciting stepping stone towards", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 433, 505, 445 ], "spans": [ { "bbox": [ 105, 433, 505, 445 ], "score": 1.0, "content": "solving more challenging mathematical problems. See, aopsbook_v2_c8_ex1, amc12b_2020_p6", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 444, 356, 455 ], "spans": [ { "bbox": [ 106, 444, 356, 455 ], "score": 1.0, "content": "and mathd_train_algebra_217 for examples of such proofs.", "type": "text" } ], "index": 28 } ], "index": 24.5 }, { "type": "text", "bbox": [ 107, 460, 505, 549 ], "lines": [ { "bbox": [ 105, 460, 506, 473 ], "spans": [ { "bbox": [ 105, 460, 506, 473 ], "score": 1.0, "content": "More generally, we also observe that proofs generated by our models have a distinctive style compared", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 471, 506, 484 ], "spans": [ { "bbox": [ 105, 471, 506, 484 ], "score": 1.0, "content": "to proofs formalized by humans. This stems in part from the model’s capability to leverage high-level", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 482, 506, 495 ], "spans": [ { "bbox": [ 105, 482, 506, 495 ], "score": 1.0, "content": "tactics in a way that is challenging for humans as discussed in this section (e.g. one-liners such", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 493, 505, 506 ], "spans": [ { "bbox": [ 105, 493, 227, 506 ], "score": 1.0, "content": "as nlinarith [sq_nonneg", "type": "text" }, { "bbox": [ 227, 494, 264, 505 ], "score": 0.8, "content": "( \\textsf { x } \\texttt { - y } )", "type": "inline_equation" }, { "bbox": [ 264, 493, 327, 506 ], "score": 1.0, "content": ", sq_nonneg", "type": "text" }, { "bbox": [ 327, 493, 371, 505 ], "score": 0.69, "content": "( \\mathsf { y } \\mathrm { ~ ~ { ~ - ~ } ~ } \\mathsf { z } ) ]", "type": "inline_equation" }, { "bbox": [ 371, 493, 505, 506 ], "score": 1.0, "content": "where humans would generally", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 504, 505, 517 ], "spans": [ { "bbox": [ 105, 504, 505, 517 ], "score": 1.0, "content": "decompose the problem in a less machine-like way). Additionally, as a result of our search procedure", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 515, 505, 528 ], "spans": [ { "bbox": [ 105, 515, 505, 528 ], "score": 1.0, "content": "and despite the bias towards shorter proofs introduced by our value function, extraneous proofsteps", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 526, 506, 539 ], "spans": [ { "bbox": [ 105, 526, 506, 539 ], "score": 1.0, "content": "(such as reversion/introduction of hypotheses, or no-op rewrites) are often interleaved with useful", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 538, 320, 549 ], "spans": [ { "bbox": [ 105, 538, 320, 549 ], "score": 1.0, "content": "ones, which rarely happens in human formalizations.", "type": "text" } ], "index": 36 } ], "index": 32.5 } ], "page_idx": 22, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 25, 293, 38 ], "spans": [ { "bbox": [ 106, 25, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 312, 763 ], "spans": [ { "bbox": [ 298, 750, 312, 763 ], "score": 1.0, "content": "23", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 106, 81, 414, 94 ], "lines": [ { "bbox": [ 105, 81, 415, 96 ], "spans": [ { "bbox": [ 105, 81, 415, 96 ], "score": 1.0, "content": "J EXAMPLE PROOFS FROM miniF2F-{test, valid, curriculum}", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 106, 504, 128 ], "lines": [ { "bbox": [ 105, 105, 506, 119 ], "spans": [ { "bbox": [ 105, 105, 506, 119 ], "score": 1.0, "content": "We present in this section proofs found by our models from miniF2F-{test, valid, curriculum},", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 117, 419, 130 ], "spans": [ { "bbox": [ 106, 117, 419, 130 ], "score": 1.0, "content": "demonstrating some of the capabilities emerging from our training procedure.", "type": "text" } ], "index": 2 } ], "index": 1.5, "bbox_fs": [ 105, 105, 506, 130 ] }, { "type": "title", "bbox": [ 108, 142, 282, 153 ], "lines": [ { "bbox": [ 105, 141, 285, 155 ], "spans": [ { "bbox": [ 105, 141, 285, 155 ], "score": 1.0, "content": "J.1 QUALITATIVE ANALYSIS OF PROOFS", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "text", "bbox": [ 107, 163, 505, 207 ], "lines": [ { "bbox": [ 106, 163, 505, 175 ], "spans": [ { "bbox": [ 106, 163, 505, 175 ], "score": 1.0, "content": "We provide qualitative insights in the nature of the proofs found by our models, which we believe", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 173, 506, 186 ], "spans": [ { "bbox": [ 105, 173, 506, 186 ], "score": 1.0, "content": "are useful to build a better intuition of their capabilities beyond pass rate numbers. Throughout this", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 185, 506, 197 ], "spans": [ { "bbox": [ 105, 185, 506, 197 ], "score": 1.0, "content": "section, we refer to statements and solutions found by our models that are presented in Appendix J", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 196, 356, 209 ], "spans": [ { "bbox": [ 105, 196, 356, 209 ], "score": 1.0, "content": "along with comments describing the specificity of each proof.", "type": "text" } ], "index": 7 } ], "index": 5.5, "bbox_fs": [ 105, 163, 506, 209 ] }, { "type": "text", "bbox": [ 107, 212, 505, 257 ], "lines": [ { "bbox": [ 105, 212, 505, 225 ], "spans": [ { "bbox": [ 105, 212, 505, 225 ], "score": 1.0, "content": "First, we observe that a large number of olympiad problems that are designed to be computationally", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 224, 505, 236 ], "spans": [ { "bbox": [ 106, 224, 505, 236 ], "score": 1.0, "content": "challenging for humans are rendered trivial for our models through the use of Lean tactics. As an", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 234, 507, 248 ], "spans": [ { "bbox": [ 105, 234, 507, 248 ], "score": 1.0, "content": "example, mathd_numbertheory_447 which is not necessarily considered straightforward for humans,", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 246, 384, 257 ], "spans": [ { "bbox": [ 105, 246, 384, 257 ], "score": 1.0, "content": "can be closed in Lean by a simple refl (proof found by our models).", "type": "text" } ], "index": 11 } ], "index": 9.5, "bbox_fs": [ 105, 212, 507, 257 ] }, { "type": "text", "bbox": [ 107, 262, 505, 361 ], "lines": [ { "bbox": [ 106, 262, 505, 274 ], "spans": [ { "bbox": [ 106, 262, 505, 274 ], "score": 1.0, "content": "In recent years, Lean’s mathlib community has developed high-powered tactics such as", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 273, 506, 286 ], "spans": [ { "bbox": [ 106, 273, 506, 286 ], "score": 1.0, "content": "linarith/nlinarith (solves (non)linear inequalities), norm_num (normalizes numerical expres-", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 285, 505, 297 ], "spans": [ { "bbox": [ 106, 285, 505, 297 ], "score": 1.0, "content": "sions), simp (simplifies goals and hypotheses) and ring (normalizes expressions in a ring). These", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 295, 504, 307 ], "spans": [ { "bbox": [ 106, 295, 504, 307 ], "score": 1.0, "content": "tactics can be used with arguments to guide their underlying search procedure. As mentioned in", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 306, 506, 319 ], "spans": [ { "bbox": [ 105, 306, 506, 319 ], "score": 1.0, "content": "Zheng et al. (2022), we confirm here that our models acquire advanced capabilities to leverage these", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 317, 505, 330 ], "spans": [ { "bbox": [ 105, 317, 505, 330 ], "score": 1.0, "content": "high-level tactics by providing exogenous arguments which are not present in the current tactic", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 327, 506, 342 ], "spans": [ { "bbox": [ 105, 327, 506, 342 ], "score": 1.0, "content": "state. The generation of these exogenous arguments through language modeling seems to require a", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 338, 505, 352 ], "spans": [ { "bbox": [ 105, 338, 505, 352 ], "score": 1.0, "content": "non-trivial amount of mathematical intuition. imo_1964_p2, imo_1961_p1 and aime_1990_p15 are", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 350, 223, 363 ], "spans": [ { "bbox": [ 105, 350, 223, 363 ], "score": 1.0, "content": "good examples of such uses.", "type": "text" } ], "index": 20 } ], "index": 16, "bbox_fs": [ 105, 262, 506, 363 ] }, { "type": "text", "bbox": [ 107, 367, 505, 455 ], "lines": [ { "bbox": [ 105, 366, 505, 379 ], "spans": [ { "bbox": [ 105, 366, 505, 379 ], "score": 1.0, "content": "We have also observed a number of proofs that require multiple non-trivial reasoning steps through the", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 378, 506, 390 ], "spans": [ { "bbox": [ 105, 378, 506, 390 ], "score": 1.0, "content": "use of lower-level tactics such as use, have, or by_cases that generally involve producing a witness or", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 388, 506, 402 ], "spans": [ { "bbox": [ 105, 388, 506, 402 ], "score": 1.0, "content": "chaining implications, requiring the generation of context specific exogenous terms. These interesting", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 399, 505, 413 ], "spans": [ { "bbox": [ 105, 399, 505, 413 ], "score": 1.0, "content": "reasoning steps are structurally different from simple normalization, simplification and rewriting", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 411, 505, 423 ], "spans": [ { "bbox": [ 105, 411, 505, 423 ], "score": 1.0, "content": "of hypotheses or goals because they heavily rely on our models ability to generate meaningful", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 421, 506, 434 ], "spans": [ { "bbox": [ 105, 421, 506, 434 ], "score": 1.0, "content": "cuts or witnesses. This capability is, in our opinion, the most exciting stepping stone towards", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 433, 505, 445 ], "spans": [ { "bbox": [ 105, 433, 505, 445 ], "score": 1.0, "content": "solving more challenging mathematical problems. See, aopsbook_v2_c8_ex1, amc12b_2020_p6", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 444, 356, 455 ], "spans": [ { "bbox": [ 106, 444, 356, 455 ], "score": 1.0, "content": "and mathd_train_algebra_217 for examples of such proofs.", "type": "text" } ], "index": 28 } ], "index": 24.5, "bbox_fs": [ 105, 366, 506, 455 ] }, { "type": "text", "bbox": [ 107, 460, 505, 549 ], "lines": [ { "bbox": [ 105, 460, 506, 473 ], "spans": [ { "bbox": [ 105, 460, 506, 473 ], "score": 1.0, "content": "More generally, we also observe that proofs generated by our models have a distinctive style compared", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 471, 506, 484 ], "spans": [ { "bbox": [ 105, 471, 506, 484 ], "score": 1.0, "content": "to proofs formalized by humans. This stems in part from the model’s capability to leverage high-level", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 482, 506, 495 ], "spans": [ { "bbox": [ 105, 482, 506, 495 ], "score": 1.0, "content": "tactics in a way that is challenging for humans as discussed in this section (e.g. one-liners such", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 493, 505, 506 ], "spans": [ { "bbox": [ 105, 493, 227, 506 ], "score": 1.0, "content": "as nlinarith [sq_nonneg", "type": "text" }, { "bbox": [ 227, 494, 264, 505 ], "score": 0.8, "content": "( \\textsf { x } \\texttt { - y } )", "type": "inline_equation" }, { "bbox": [ 264, 493, 327, 506 ], "score": 1.0, "content": ", sq_nonneg", "type": "text" }, { "bbox": [ 327, 493, 371, 505 ], "score": 0.69, "content": "( \\mathsf { y } \\mathrm { ~ ~ { ~ - ~ } ~ } \\mathsf { z } ) ]", "type": "inline_equation" }, { "bbox": [ 371, 493, 505, 506 ], "score": 1.0, "content": "where humans would generally", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 504, 505, 517 ], "spans": [ { "bbox": [ 105, 504, 505, 517 ], "score": 1.0, "content": "decompose the problem in a less machine-like way). Additionally, as a result of our search procedure", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 515, 505, 528 ], "spans": [ { "bbox": [ 105, 515, 505, 528 ], "score": 1.0, "content": "and despite the bias towards shorter proofs introduced by our value function, extraneous proofsteps", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 526, 506, 539 ], "spans": [ { "bbox": [ 105, 526, 506, 539 ], "score": 1.0, "content": "(such as reversion/introduction of hypotheses, or no-op rewrites) are often interleaved with useful", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 538, 320, 549 ], "spans": [ { "bbox": [ 105, 538, 320, 549 ], "score": 1.0, "content": "ones, which rarely happens in human formalizations.", "type": "text" } ], "index": 36 } ], "index": 32.5, "bbox_fs": [ 105, 460, 506, 549 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 83, 162, 94 ], "lines": [ { "bbox": [ 106, 81, 163, 95 ], "spans": [ { "bbox": [ 106, 81, 163, 95 ], "score": 1.0, "content": "imo_1961_p1", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table", "bbox": [ 106, 100, 501, 570 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 100, 501, 570 ], "group_id": 0, "lines": [ { "bbox": [ 106, 100, 501, 570 ], "spans": [ { "bbox": [ 106, 100, 501, 570 ], "score": 0.981, "html": "
Natural languageSolve the system of equations:x+y+z=αx²+y²+2²=b²xy= x²where α and b are constants.Give the conditions that α and b must satisfy sothat x,y,z (the solutions of the system) are distinct positive numbers.Note: theformalized statement in miniF2F is a weaker problem as it focuses on the secondpart of the question, providing the actual conditions,and asking for a proof that therequirement entails them.
Model prooftheorem imo_1961_p1(xyzab:R)(ho:0<x>0<y>0<z)(h1:×≠y)(h2 :y≠z)(h3:z≠x)(h4 :x+y+z= a)(h5 :x^2 +y^2 + z^2 = b^2)(h6 :x *y= z^2) :0<a∧ b^2<a^2 > a^2<3*b^2 :=beginrevert_all,intros,rw mul_comm,split,{nlinarith [sq_nonneg (x - y)，sq_nonneg(y- z)]，},split,{nlinarith [sq_nonneg(z - 1)]，},revert h3 h4,field_simp [mul_comm a b],rw [mul_comm，← h5],contrapose!,rw mul_comm at h6,rw mul_comm,intro h,nlinarith [sq_nonneg (x - y)，sq_nonneg (y - z)]end
CommentsThe model is able to close this problem by spliting into cases,contraposing for thelast case and using nlinarith.It must be noted that the arguments for the first twonlinarith uses are not necessary,however the [sq_nonneg (x - y)，sq_nonneg(y- z)] argument provided on the last line is crucial to close the goal and arecompletely exogenous (present in no form in the tactic state before).
", "type": "table", "image_path": "114b1758b4db4a1da652cf833c9fe05ab5dc0caf16d6575d5695ba11c16559fe.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 100, 501, 256.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 106, 256.66666666666663, 501, 413.33333333333326 ], "spans": [], "index": 2 }, { "bbox": [ 106, 413.33333333333326, 501, 569.9999999999999 ], "spans": [], "index": 3 } ] } ], "index": 2 } ], "page_idx": 23, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 764 ], "spans": [ { "bbox": [ 298, 750, 313, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 15 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 83, 162, 94 ], "lines": [ { "bbox": [ 106, 81, 163, 95 ], "spans": [ { "bbox": [ 106, 81, 163, 95 ], "score": 1.0, "content": "imo_1961_p1", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 82, 164, 95 ], "spans": [ { "bbox": [ 106, 82, 164, 95 ], "score": 1.0, "content": "imo_1964_p2", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 82, 174, 95 ], "spans": [ { "bbox": [ 106, 82, 174, 95 ], "score": 1.0, "content": "aime_1990_p15", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 83, 223, 94 ], "spans": [ { "bbox": [ 106, 83, 223, 94 ], "score": 1.0, "content": "mathd_train_algebra_217", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 82, 179, 95 ], "spans": [ { "bbox": [ 106, 82, 179, 95 ], "score": 1.0, "content": "amc12b_2020_p6", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 82, 193, 95 ], "spans": [ { "bbox": [ 106, 82, 193, 95 ], "score": 1.0, "content": "mathd_algebra_140", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 82, 168, 95 ], "spans": [ { "bbox": [ 106, 82, 168, 95 ], "score": 1.0, "content": "aime_1984_p1", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 82, 198, 96 ], "spans": [ { "bbox": [ 106, 82, 198, 96 ], "score": 1.0, "content": "aopsbook_v2_c8_ex1", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 106, 495, 218, 507 ], "spans": [ { "bbox": [ 106, 495, 218, 507 ], "score": 1.0, "content": "mathd_numbertheory_447", "type": "text", "cross_page": true } ], "index": 4 } ], "index": 0, "bbox_fs": [ 106, 81, 163, 95 ] }, { "type": "table", "bbox": [ 106, 100, 501, 570 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 100, 501, 570 ], "group_id": 0, "lines": [ { "bbox": [ 106, 100, 501, 570 ], "spans": [ { "bbox": [ 106, 100, 501, 570 ], "score": 0.981, "html": "
Natural languageSolve the system of equations:x+y+z=αx²+y²+2²=b²xy= x²where α and b are constants.Give the conditions that α and b must satisfy sothat x,y,z (the solutions of the system) are distinct positive numbers.Note: theformalized statement in miniF2F is a weaker problem as it focuses on the secondpart of the question, providing the actual conditions,and asking for a proof that therequirement entails them.
Model prooftheorem imo_1961_p1(xyzab:R)(ho:0<x>0<y>0<z)(h1:×≠y)(h2 :y≠z)(h3:z≠x)(h4 :x+y+z= a)(h5 :x^2 +y^2 + z^2 = b^2)(h6 :x *y= z^2) :0<a∧ b^2<a^2 > a^2<3*b^2 :=beginrevert_all,intros,rw mul_comm,split,{nlinarith [sq_nonneg (x - y)，sq_nonneg(y- z)]，},split,{nlinarith [sq_nonneg(z - 1)]，},revert h3 h4,field_simp [mul_comm a b],rw [mul_comm，← h5],contrapose!,rw mul_comm at h6,rw mul_comm,intro h,nlinarith [sq_nonneg (x - y)，sq_nonneg (y - z)]end
CommentsThe model is able to close this problem by spliting into cases,contraposing for thelast case and using nlinarith.It must be noted that the arguments for the first twonlinarith uses are not necessary,however the [sq_nonneg (x - y)，sq_nonneg(y- z)] argument provided on the last line is crucial to close the goal and arecompletely exogenous (present in no form in the tactic state before).
", "type": "table", "image_path": "114b1758b4db4a1da652cf833c9fe05ab5dc0caf16d6575d5695ba11c16559fe.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 100, 501, 256.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 106, 256.66666666666663, 501, 413.33333333333326 ], "spans": [], "index": 2 }, { "bbox": [ 106, 413.33333333333326, 501, 569.9999999999999 ], "spans": [], "index": 3 } ] } ], "index": 2 } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 83, 163, 94 ], "lines": [ { "bbox": [ 106, 82, 164, 95 ], "spans": [ { "bbox": [ 106, 82, 164, 95 ], "score": 1.0, "content": "imo_1964_p2", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table", "bbox": [ 106, 107, 503, 345 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 107, 503, 345 ], "group_id": 0, "lines": [ { "bbox": [ 106, 107, 503, 345 ], "spans": [ { "bbox": [ 106, 107, 503, 345 ], "score": 0.983, "html": "
Natural languageSuppose a,b,c are the sides of a triangle.Prove thata²(b+c-a)+b²(c+a-b)+c²(a+b-c)≤3abc
Model prooftheorem imo_1964_p2(abc:R)(ho:0<a>0<b>0<c)(h1:c<a+b)(h2:b<a+c)(h3:a<b+c):a^2*(b+c-a)+b^2*(c+a-b)+c^2*(a+b-c)≤3*a*b*c:=beginnlinarith [sq_nonneg (b - a)，sq_nonneg (c - b),sq_nonneg (a - c)，sq_nonneg (c - a)]end
CommentsThe model is able to close an IMO problem in one-line.It correctly providesexogenous arguments to nlinarith,which are necessary to close the goal. Notethat either one of the last two arguments in the sequence [sq_nonneg (b - a),sq_nonneg(c -b)，sq_nonneg(a - c)，sq_nonneg (c- a)]can be omitted.
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Natural languageSuppose a,b,c are the sides of a triangle.Prove thata²(b+c-a)+b²(c+a-b)+c²(a+b-c)≤3abc
Model prooftheorem imo_1964_p2(abc:R)(ho:0<a>0<b>0<c)(h1:c<a+b)(h2:b<a+c)(h3:a<b+c):a^2*(b+c-a)+b^2*(c+a-b)+c^2*(a+b-c)≤3*a*b*c:=beginnlinarith [sq_nonneg (b - a)，sq_nonneg (c - b),sq_nonneg (a - c)，sq_nonneg (c - a)]end
CommentsThe model is able to close an IMO problem in one-line.It correctly providesexogenous arguments to nlinarith,which are necessary to close the goal. Notethat either one of the last two arguments in the sequence [sq_nonneg (b - a),sq_nonneg(c -b)，sq_nonneg(a - c)，sq_nonneg (c- a)]can be omitted.
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Natural languageFind ax+ by if the real numbers a,b,x,and y satisfy the equations ax+by =3, ax²+by²=7, ax²+by³=16, ax²4 +by4 = 42. Note: the formalized statement in miniF2F provides the answer and asks for a proof of it. theorem aime_1990_p15
Model proof(abxy:R) (ho:a*x+b*y=3) (h1 :a* x^2+b*y^2= 7) (h2 :a* x^3 +b*y^3= 16) (h3 :a* x^4 +b*y^4= 42) : a * x^5+b*y^5=20 := begin revert_all, intros ab ×y hg hi h2 h4, ring_nf at hi h2, rw ← sub_eq_zero at h1, nlinarith [sq_nonneg (× - y)，sq_nonneg (a + b - 2), sq_nonneg (x + y - (2:R))，sq_nonneg (a -b - 2)]
Commentsend The model is able to close a challenging AIME problem by providing crucial ex- ogenous arguments sq_nonneg (x - y) and sq_nonneg (x + y - (2 :R)) to nlinarith,which are required to close the goal (while the other two can be removed).
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Natural languageFind ax+ by if the real numbers a,b,x,and y satisfy the equations ax+by =3, ax²+by²=7, ax²+by³=16, ax²4 +by4 = 42. Note: the formalized statement in miniF2F provides the answer and asks for a proof of it. theorem aime_1990_p15
Model proof(abxy:R) (ho:a*x+b*y=3) (h1 :a* x^2+b*y^2= 7) (h2 :a* x^3 +b*y^3= 16) (h3 :a* x^4 +b*y^4= 42) : a * x^5+b*y^5=20 := begin revert_all, intros ab ×y hg hi h2 h4, ring_nf at hi h2, rw ← sub_eq_zero at h1, nlinarith [sq_nonneg (× - y)，sq_nonneg (a + b - 2), sq_nonneg (x + y - (2:R))，sq_nonneg (a -b - 2)]
Commentsend The model is able to close a challenging AIME problem by providing crucial ex- ogenous arguments sq_nonneg (x - y) and sq_nonneg (x + y - (2 :R)) to nlinarith,which are required to close the goal (while the other two can be removed).
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Natural languageLet f(x)= Ax+B and g(x)=Bx +A,where A≠B. If f(g(x)) - g(f(x))= B-A,what is A+B? Note: the formalized statement in our curriculum provides the answer and asks for a proof of it. theorem mathd_train_algebra_217
Model proof(ab:R) (fg:R→R) (ho:∀x，fx=a*x+b) (h1:∀×，fx=b*x+a) (h2 :a≠b) (h3 :∀x，f (gx)-g(f x)=b-a): a+b=0 := begin revert_all, intros a b, intros f g, contrapose!, rintro <ho，<hi，h2>>, use (0 :R), simp only[sub_eq_iff_eq_add，ho，mul_zero]at *, simp only[*，zero_add],
Commentsnorm_num at ho end The model is able to close the goal by contraposing,supplying a witness by the use of use (O :R）and finally leveraging the simp and norm_num. This example demonstrates the model's ability to chain multiple non-trivial steps of reasoning including the generation of witnesses.
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Natural languageLet f(x)= Ax+B and g(x)=Bx +A,where A≠B. If f(g(x)) - g(f(x))= B-A,what is A+B? Note: the formalized statement in our curriculum provides the answer and asks for a proof of it. theorem mathd_train_algebra_217
Model proof(ab:R) (fg:R→R) (ho:∀x，fx=a*x+b) (h1:∀×，fx=b*x+a) (h2 :a≠b) (h3 :∀x，f (gx)-g(f x)=b-a): a+b=0 := begin revert_all, intros a b, intros f g, contrapose!, rintro <ho，<hi，h2>>, use (0 :R), simp only[sub_eq_iff_eq_add，ho，mul_zero]at *, simp only[*，zero_add],
Commentsnorm_num at ho end The model is able to close the goal by contraposing,supplying a witness by the use of use (O :R）and finally leveraging the simp and norm_num. This example demonstrates the model's ability to chain multiple non-trivial steps of reasoning including the generation of witnesses.
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Natural language (A) a multiple of 4 (D) a perfect squareFor all integers n ≥ 9, the value of (n+2)!-(n+1)!
n! is always which of the following? (B) a multiple of 10 (C)a prime number
Model prooftheorem amc12b_2020_p6 (n :N) (ho:9≤n): x :N，(x:R)^2 = (nat.factorial (n + 2) - nat.factorial (n + 1)）/ nat.factorial n := begin revert_all, intros n hn, use n + 1,field_simp [nat.factorial_ne_zero，pow_succ'],
Commentsring_exp end The model exhibits here its ability to implicitly interpret factorial expressions by guessing the correct answer with the use of use n + 1 and straightforwardly proving
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Natural language (A) a multiple of 4 (D) a perfect squareFor all integers n ≥ 9, the value of (n+2)!-(n+1)!
n! is always which of the following? (B) a multiple of 10 (C)a prime number
Model prooftheorem amc12b_2020_p6 (n :N) (ho:9≤n): x :N，(x:R)^2 = (nat.factorial (n + 2) - nat.factorial (n + 1)）/ nat.factorial n := begin revert_all, intros n hn, use n + 1,field_simp [nat.factorial_ne_zero，pow_succ'],
Commentsring_exp end The model exhibits here its ability to implicitly interpret factorial expressions by guessing the correct answer with the use of use n + 1 and straightforwardly proving
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Natural languageThe expression 24x² - 19x - 35 can be written as (Ax - 5)(2Bx + C),where A, B,and C are positive numbers.Find AB- 3C.Note: the formalized statement in miniF2F provides the answer and asks for a proof of it.
Model prooftheorem mathd_algebra_140 (abc:R) (ho:0<a>0<b>0<c) (h1:∀×，24* ×^2-19*×-35= (((a*x)-5)*((2*(b*x))+c))）: a*b-3*c=-9:= begin revert_all, rintro abc h1, rw mul_comm, rw ←sub_eq_zero, field_simp, rw sub_eq_add_neg, rw←eq_neg_iff_add_eq_zero, rw [mul_comm，← sub_eq_add_neg，sub_eq_add_neg, ← sub_eq_add_neg], rw ←sub_eq_zero, simp only [mul_add，neg_mul_eq_neg_mul_symm，mul_comm, add_left_comm], norm_num, simp only [pow_two], intro ho, rw← sub_eq_zero, linarith [hi.2，hi.1，ho 3，ho 1，ho 2]
Commentsend The model mostly "struggles" to make progress up to the last line. There, it presents us with the idea to specialize the statement given in h1 (which became ho in the process of the proof) at three consecutive natural numbers 1,2,3 which closes the goal with nlinarith. This proof is interesting as it demonstrates the model's ability to evaluate symbolic expressions implicitly.
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Natural languageThe expression 24x² - 19x - 35 can be written as (Ax - 5)(2Bx + C),where A, B,and C are positive numbers.Find AB- 3C.Note: the formalized statement in miniF2F provides the answer and asks for a proof of it.
Model prooftheorem mathd_algebra_140 (abc:R) (ho:0<a>0<b>0<c) (h1:∀×，24* ×^2-19*×-35= (((a*x)-5)*((2*(b*x))+c))）: a*b-3*c=-9:= begin revert_all, rintro abc h1, rw mul_comm, rw ←sub_eq_zero, field_simp, rw sub_eq_add_neg, rw←eq_neg_iff_add_eq_zero, rw [mul_comm，← sub_eq_add_neg，sub_eq_add_neg, ← sub_eq_add_neg], rw ←sub_eq_zero, simp only [mul_add，neg_mul_eq_neg_mul_symm，mul_comm, add_left_comm], norm_num, simp only [pow_two], intro ho, rw← sub_eq_zero, linarith [hi.2，hi.1，ho 3，ho 1，ho 2]
Commentsend The model mostly "struggles" to make progress up to the last line. There, it presents us with the idea to specialize the statement given in h1 (which became ho in the process of the proof) at three consecutive natural numbers 1,2,3 which closes the goal with nlinarith. This proof is interesting as it demonstrates the model's ability to evaluate symbolic expressions implicitly.
", "type": "table", "image_path": "72e180128e967524d6ab2f44b2e5633e8ef7b3b5c98aca7e96b6feff7a9c7631.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 107, 502, 226.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 106, 226.66666666666669, 502, 346.33333333333337 ], "spans": [], "index": 2 }, { "bbox": [ 106, 346.33333333333337, 502, 466.00000000000006 ], "spans": [], "index": 3 } ] } ], "index": 2 } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 83, 167, 94 ], "lines": [ { "bbox": [ 106, 82, 168, 95 ], "spans": [ { "bbox": [ 106, 82, 168, 95 ], "score": 1.0, "content": "aime_1984_p1", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table", "bbox": [ 106, 108, 502, 383 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 108, 502, 383 ], "group_id": 0, "lines": [ { "bbox": [ 106, 108, 502, 383 ], "spans": [ { "bbox": [ 106, 108, 502, 383 ], "score": 0.981, "html": "
Natural languageFind the value of a2 +a4 +a6 +ag +...+ ag if a1,a2,a3... is an arithmetic progression with common difference 1,and a1 +a2 + a3 +...+ag8 = 137.Note: the formalized statement in miniF2F provides the answer and asks for a proof of it.
Model prooftheorem aime_1984_p1 (u:N→Q) (ho :∀n，u(n+1)=un+1) (hi ：∑k in finset.range 98，u k.succ = 137）: ∑ k in finset.range 49，u (2 * k.succ)= 93 := begin revert_all, simp [finset.sum_range_succ], ring, simp[pow_succ，mul_comm，add_left_comm，add_assoc], simp [two_mul], rintro p, revert p, simp [add_comm], intros p hp, simp [hp]，ring_nf, intros, nlinarith
Commentsend The model finds a rather short proof of the statement compared to typ- ical human formalizations (see https://gist.github.com/MantasBaksys/ 3efd5c0d42f440d16c09076db42822f2).
", "type": "table", "image_path": "3208d93002bdd4ce9911a6acba8842461d264cb01ea72da3584f72edd2c73ff1.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 108, 502, 199.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 106, 199.66666666666669, 502, 291.33333333333337 ], "spans": [], "index": 2 }, { "bbox": [ 106, 291.33333333333337, 502, 383.00000000000006 ], "spans": [], "index": 3 } ] } ], "index": 2 } ], "page_idx": 29, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 763 ], "spans": [ { "bbox": [ 298, 750, 313, 763 ], "score": 1.0, "content": "30", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 83, 167, 94 ], "lines": [], "index": 0, "bbox_fs": [ 106, 82, 168, 95 ], "lines_deleted": true }, { "type": "table", "bbox": [ 106, 108, 502, 383 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 108, 502, 383 ], "group_id": 0, "lines": [ { "bbox": [ 106, 108, 502, 383 ], "spans": [ { "bbox": [ 106, 108, 502, 383 ], "score": 0.981, "html": "
Natural languageFind the value of a2 +a4 +a6 +ag +...+ ag if a1,a2,a3... is an arithmetic progression with common difference 1,and a1 +a2 + a3 +...+ag8 = 137.Note: the formalized statement in miniF2F provides the answer and asks for a proof of it.
Model prooftheorem aime_1984_p1 (u:N→Q) (ho :∀n，u(n+1)=un+1) (hi ：∑k in finset.range 98，u k.succ = 137）: ∑ k in finset.range 49，u (2 * k.succ)= 93 := begin revert_all, simp [finset.sum_range_succ], ring, simp[pow_succ，mul_comm，add_left_comm，add_assoc], simp [two_mul], rintro p, revert p, simp [add_comm], intros p hp, simp [hp]，ring_nf, intros, nlinarith
Commentsend The model finds a rather short proof of the statement compared to typ- ical human formalizations (see https://gist.github.com/MantasBaksys/ 3efd5c0d42f440d16c09076db42822f2).
", "type": "table", "image_path": "3208d93002bdd4ce9911a6acba8842461d264cb01ea72da3584f72edd2c73ff1.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 108, 502, 199.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 106, 199.66666666666669, 502, 291.33333333333337 ], "spans": [], "index": 2 }, { "bbox": [ 106, 291.33333333333337, 502, 383.00000000000006 ], "spans": [], "index": 3 } ] } ], "index": 2 } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 83, 197, 94 ], "lines": [ { "bbox": [ 106, 82, 198, 96 ], "spans": [ { "bbox": [ 106, 82, 198, 96 ], "score": 1.0, "content": "aopsbook_v2_c8_ex1", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table", "bbox": [ 106, 105, 502, 475 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 105, 502, 475 ], "group_id": 0, "lines": [ { "bbox": [ 106, 105, 502, 475 ], "spans": [ { "bbox": [ 106, 105, 502, 475 ], "score": 0.982, "html": "
Natural languageConsider the sequence = 1 3 ，，，. Prove that this sequence tends to 1 as n→
Model prooftheorem aopsbook_v2_c8_ex1 (u:N→R) (ho:∀n，un=n/(n+1)): filter.tendsto u filter.at_top (N 1) := begin revert_all, simp [← nnreal.coe_one], norm_cast, intros, revert ho, assume h, simp [tendsto_const_nhds，← nnreal.coe_one，h], revert u, assume f, norm_num, rw tendsto_iff_norm_tendsto_zero, assume H, convert tendsto_norm_zero.comp tendsto_one_div_add_at_top_nhds_0_nat, funext n, have ho ：（((n:R）+1）:R)≠0, { norm_cast，exact n.succ_ne_zero，}, rwH, field_simp [ho，norm_neg], ring, rw [← sub_eq_zero], simp [← sub_eq_zero], simp[sub_eq_add_neg],
Commentsend An interesting example of a generated have statement,denoted by ho,which is introduced as a cut in order to simplify the expression containing divisions by using field_simp[ho，norm_neg] ata later step.
", "type": "table", "image_path": "63d1a900dd3a44c7dd5046e3d4d74f93c7d77df71fdeb0a2563d1255d7d889c0.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 105, 502, 228.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 228.33333333333331, 502, 351.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 351.66666666666663, 502, 474.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 2 }, { "type": "text", "bbox": [ 106, 495, 217, 506 ], "lines": [ { "bbox": [ 106, 495, 218, 507 ], "spans": [ { "bbox": [ 106, 495, 218, 507 ], "score": 1.0, "content": "mathd_numbertheory_447", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "table", "bbox": [ 106, 520, 502, 657 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 520, 502, 657 ], "group_id": 1, "lines": [ { "bbox": [ 106, 520, 502, 657 ], "spans": [ { "bbox": [ 106, 520, 502, 657 ], "score": 0.982, "html": "
Natural languageWhat is the sum of the units digits of all the multiples of 3 between O and 5O? Note:the formalized statement in miniF2F provides the answer and asks for a proof of it.
Model prooftheorem mathd_numbertheory_447 :∑ k in finset.filter (入 ×，3|x)(finset.erase (finset.range 50) 0)，(k % 10) = 78 :=beginreflend
CommentsBecause the predicate 入 ×，3|× is registered as decidable over N,we can state theproblem by using finset.filter,which is computable.Hence,refl is able toclose the goal.
", "type": "table", "image_path": "306327dca32d89585f5ece28b86e199dbe353bbe99bf626ab435d3110445f844.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 106, 520, 502, 565.6666666666666 ], "spans": [], "index": 5 }, { "bbox": [ 106, 565.6666666666666, 502, 611.3333333333333 ], "spans": [], "index": 6 }, { "bbox": [ 106, 611.3333333333333, 502, 656.9999999999999 ], "spans": [], "index": 7 } ] } ], "index": 6 } ], "page_idx": 30, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 293, 37 ], "lines": [ { "bbox": [ 106, 26, 293, 38 ], "spans": [ { "bbox": [ 106, 26, 293, 38 ], "score": 1.0, "content": "Published as a conference paper at ICLR 2023", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 83, 197, 94 ], "lines": [], "index": 0, "bbox_fs": [ 106, 82, 198, 96 ], "lines_deleted": true }, { "type": "table", "bbox": [ 106, 105, 502, 475 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 105, 502, 475 ], "group_id": 0, "lines": [ { "bbox": [ 106, 105, 502, 475 ], "spans": [ { "bbox": [ 106, 105, 502, 475 ], "score": 0.982, "html": "
Natural languageConsider the sequence = 1 3 ，，，. Prove that this sequence tends to 1 as n→
Model prooftheorem aopsbook_v2_c8_ex1 (u:N→R) (ho:∀n，un=n/(n+1)): filter.tendsto u filter.at_top (N 1) := begin revert_all, simp [← nnreal.coe_one], norm_cast, intros, revert ho, assume h, simp [tendsto_const_nhds，← nnreal.coe_one，h], revert u, assume f, norm_num, rw tendsto_iff_norm_tendsto_zero, assume H, convert tendsto_norm_zero.comp tendsto_one_div_add_at_top_nhds_0_nat, funext n, have ho ：（((n:R）+1）:R)≠0, { norm_cast，exact n.succ_ne_zero，}, rwH, field_simp [ho，norm_neg], ring, rw [← sub_eq_zero], simp [← sub_eq_zero], simp[sub_eq_add_neg],
Commentsend An interesting example of a generated have statement,denoted by ho,which is introduced as a cut in order to simplify the expression containing divisions by using field_simp[ho，norm_neg] ata later step.
", "type": "table", "image_path": "63d1a900dd3a44c7dd5046e3d4d74f93c7d77df71fdeb0a2563d1255d7d889c0.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 105, 502, 228.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 106, 228.33333333333331, 502, 351.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 351.66666666666663, 502, 474.99999999999994 ], "spans": [], "index": 3 } ] } ], "index": 2 }, { "type": "text", "bbox": [ 106, 495, 217, 506 ], "lines": [], "index": 4, "bbox_fs": [ 106, 495, 218, 507 ], "lines_deleted": true }, { "type": "table", "bbox": [ 106, 520, 502, 657 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 520, 502, 657 ], "group_id": 1, "lines": [ { "bbox": [ 106, 520, 502, 657 ], "spans": [ { "bbox": [ 106, 520, 502, 657 ], "score": 0.982, "html": "
Natural languageWhat is the sum of the units digits of all the multiples of 3 between O and 5O? Note:the formalized statement in miniF2F provides the answer and asks for a proof of it.
Model prooftheorem mathd_numbertheory_447 :∑ k in finset.filter (入 ×，3|x)(finset.erase (finset.range 50) 0)，(k % 10) = 78 :=beginreflend
CommentsBecause the predicate 入 ×，3|× is registered as decidable over N,we can state theproblem by using finset.filter,which is computable.Hence,refl is able toclose the goal.
", "type": "table", "image_path": "306327dca32d89585f5ece28b86e199dbe353bbe99bf626ab435d3110445f844.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 106, 520, 502, 565.6666666666666 ], "spans": [], "index": 5 }, { "bbox": [ 106, 565.6666666666666, 502, 611.3333333333333 ], "spans": [], "index": 6 }, { "bbox": [ 106, 611.3333333333333, 502, 656.9999999999999 ], "spans": [], "index": 7 } ] } ], "index": 6 } ] } ], "_backend": "pipeline", "_version_name": "2.2.2" }