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Three embeddings of FML into classical higher-order logic (HOL) are provided side by side: a deep embedding, a heavyweight maximal-shallow embedding, and a lightweight minimal-shallow embedding. The minimal-shallow embedding is presented as an Isabelle/HOL locale, parametrised by an accessibility relation, a world-indexed interpretation, a universe of worlds, and a variable assignment; the locale form admits a global faithfulness theorem, stating that quantifying over all minimal-shallow interpretations recovers exactly deep validity.\n  A central technical contribution is a mechanisation, for FML under constant-domain Kripke semantics, of the (countable) downward L\\\"owenheim-Skolem theorem, which underpins the automation of our faithfulness proof between th","title":"First-Order Modal Logic in HOL: Deep and Shallow Embeddings with Automated Faithfulness (Extended Preprint)","url":"https://arxiv.org/abs/2607.10880","vendor":"arxiv_cs_ai"},"summary":"arXiv:2607.10880v2 Announce Type: replace \nAbstract: We extend, in Isabelle/HOL, the deep-and-shallow embedding methodology of our prior work from propositional to first-order modal logic (FML) with constant-domain Kripke semantics. Three embeddings of FML into classical higher-order logic (HOL) are provided side by side: a deep embedding, a heavyweight maximal-shallow embedding, and a lightweight minimal-shallow embedding. The minimal-shallow embedding is presented as an Isabelle/HOL locale, parametrised by an accessibility relation, a world-indexed interpretation, a universe of worlds, and a variable assignment; the locale form admits a global faithfulness theorem, stating that quantifying over all minimal-shallow interpretations recovers exactly deep validity.\n  A central technical contribution is a mechanisation, for FML under constant-domain Kripke semantics, of the (countable) downward L\\\"owenheim-Skolem theorem, which underpins the automation of our faithfulness proof between th","title":"First-Order Modal Logic in HOL: Deep and Shallow Embeddings with Automated Faithfulness (Extended Preprint)","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-07-20T04:43:09Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2607.10880"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:512d9a86443768fa154f93316b25b6236bd22ae36a66ef53303ed80f01212db0043a76724cec14b59569916221d50a6066af8ca6d19d1bd114b1750f8c0e710f","signer":"crovia.substrate","subject":{"observed_at":"2026-07-20T04:43:09Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2607.10880"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"dfb552e8cfe2f642be9d821434cf2a4d62e0ee5ba6526df8f02dcc4fac9de19a","leaf_index":333353,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"475fae59d764ef085b6e5674db87bbeb110feacf5603a0f3a455e233f0f61012","side":"left"},{"sibling":"e10dbaba43ca7acd4dbeca2fd3e87cc55c0a8f0d684f2f936b258a93675f413b","side":"right"},{"sibling":"7d81c5e2555db17fe74d83d5caf72dab18ff3108974eed692cf4c81e6eeb6de7","side":"right"},{"sibling":"bc24bcef91de9358d2e0a8b094cca7d71d302b59d828477c03becee376a3bab8","side":"left"},{"sibling":"7a7893cc8f38a5f9a49eedec199c3ec5e1a71385c85563ac0cea3cb31a258d29","side":"right"},{"sibling":"0d19791e9aa074b8130eeb2e01249774582a5873ad2fb1aeac0e5267891a9a37","side":"left"},{"sibling":"335e5d86a4ea00f87721afcb0e730f04bbef2c17cd5489f5b8263047ffc4ccd3","side":"right"},{"sibling":"5f23b1a0a67d8fa1aa690680510620f249f8d6683d46c202fca306510fbe398e","side":"right"},{"sibling":"f720760992870795e6d2b913ad9162f9e144b821ea97e4dada744d0cac06e93b","side":"right"},{"sibling":"45c0e4431502711514503abd48e8b3d34ee2f4bffa994c883aaf2adecd0ad8e9","side":"left"},{"sibling":"dedd2da92d9447ddf1b1db68fe20109a426ed18359861ed746907ac820021a8d","side":"left"},{"sibling":"a1c43cc7cd9c775fac33940ee5124aece01596733f003fc53743f43483f9f597","side":"right"},{"sibling":"b5ad3eafd7eeb74c063261356fdd9bf6059ee6d0bf1e3c70e60731b394a5536e","side":"left"},{"sibling":"93e399d152203db688c6a5a58d25131205603504f5b79123a1f2b5a5ed9c1e54","side":"right"},{"sibling":"b6e0cad7f6eb9107f0edd276f1a9942635d8cd6d60d2a97e7daac08b110dc209","side":"right"},{"sibling":"80ec062e7e625dc3f9bb5865cb5198696bbec2608e48abae5670677b90695899","side":"right"},{"sibling":"77025bcb374a7ad74f520643e20a8ae1205a7ee78507b0beb93117760f1c29d3","side":"left"},{"sibling":"0aced6f0c9dec3e6cc9e89b68b70f5f8ce7e1eb13606d92db1917b76e57393c7","side":"right"},{"sibling":"1cecb7f447febd025aac272837c80de218aecc6485d2395a509b2a1f1b9c746e","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":333540,"merkle_root":"ee60f62b8a724dd9bde638d638caf32cefec4440f832018b457ff47a0ec56a8c","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260720T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-07-20T05:38:36Z","sig_algorithm":"ed25519","signature":"82a3787e628bfab19c377d875220e1aaedfc498736c545f4708a1b887e8398afdf306995994493a36c864ff7139a2d436b905ce7081aaa540789aa4f707dc800","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_042f166627581fa32bd96731e83d73f7a79b38110ecdb7ec99d90bb83f4bcd14"}}