bibkey: foundation2026firstorder authors: FormalizedFormalLogic contributors year: 2026 title: Foundation first-order logic and set theory, revision 30a16ffa doi: null url: https://github.com/FormalizedFormalLogic/Foundation/tree/30a16ffa93d79d73ab4d02427fa00f50e039bf29 claim: Licensed logic, predicate-term and finite formula syntax, entailment, finite support, language maps, empty-domain elimination and unique choice for first-order developments. strata_touched:

• D5/S3/ConceptDynamics/ZfcEntailment/CalculusOne
• D5/S3/ConceptDynamics/ZfcEntailment/CalculusTwo
• D5/S3/ConceptDynamics/ZfcEntailment/EntailmentOne
• D5/S3/ConceptDynamics/ZfcEntailment/EntailmentTwo
• D5/S3/ConceptDynamics/ZfcFiniteCollections/Finset
• D5/S3/ConceptDynamics/ZfcFiniteCollections/List
• D5/S3/ConceptDynamics/ZfcFiniteCollections/Matrix
• D5/S3/ConceptDynamics/ZfcFiniteCollections/OrderDense
• D5/S3/ConceptDynamics/ZfcFiniteCollections/Quotient
• D5/S3/ConceptDynamics/ZfcFiniteData/Fin
• D5/S3/ConceptDynamics/ZfcFiniteData/Nat
• D5/S3/ConceptDynamics/ZfcFiniteData/NatMatrix
• D5/S3/ConceptDynamics/ZfcLanguageSupport/Empty
• D5/S3/ConceptDynamics/ZfcLanguageSupport/NotationClass
• D5/S3/ConceptDynamics/ZfcLogic/ForcingRelation
• D5/S3/ConceptDynamics/ZfcLogic/LogicSymbolOne
• D5/S3/ConceptDynamics/ZfcLogic/LogicSymbolTwo
• D5/S3/ConceptDynamics/ZfcLogic/Semantics
• D5/S3/ConceptDynamics/ZfcMinimalLogic/MinimalEntailmentFour
• D5/S3/ConceptDynamics/ZfcMinimalLogic/MinimalEntailmentOne
• D5/S3/ConceptDynamics/ZfcMinimalLogic/MinimalEntailmentThree
• D5/S3/ConceptDynamics/ZfcMinimalLogic/MinimalEntailmentTwo
• D5/S3/ConceptDynamics/ZfcPredicate/LanguageOne
• D5/S3/ConceptDynamics/ZfcPredicate/LanguageTwo
• D5/S3/ConceptDynamics/ZfcPredicate/Quantifier
• D5/S3/ConceptDynamics/ZfcPredicate/Term
• D5/S3/ConceptDynamics/ZfcPropositional/ClEntailment
• D5/S3/ConceptDynamics/ZfcPropositional/IntEntailmentOne
• D5/S3/ConceptDynamics/ZfcPropositional/IntEntailmentTwo
• D5/S3/ConceptDynamics/ZfcSupport/AdjunctiveSet
• D5/S3/ConceptDynamics/ZfcSupport/Function
• D5/S3/ConceptDynamics/ZfcSupport/UniqueChoice
• D5/S3/ConceptDynamics/ZfcSyntax/FormulaOne
• D5/S3/ConceptDynamics/ZfcSyntax/FormulaTwo
• D5/S3/ConceptDynamics/ZfcTermRewriting/RewOne
• D5/S3/ConceptDynamics/ZfcTermRewriting/RewTwo
• D5/S3/ConceptDynamics/ZfcTermRewriting/RewThree
• D5/S3/ConceptDynamics/ZfcTermRewriting/RewFour license: Apache-2.0 triage: anchor

foundation2026firstorder

Verified locator

Immutable source: FormalizedFormalLogic/Foundation, revision 30a16ffa93d79d73ab4d02427fa00f50e039bf29. The table below locates the selected sources within this immutable revision; its SHA-256 digests identify the complete original upstream files. Copyright and attribution remain with the upstream contributors. Original author notices and required source notices are preserved. The upstream distribution has no NOTICE file. The complete unmodified Apache-2.0 license follows below.

The table records the Foundation source ancestry of the installed Lean paths; each strata_touched GID resolves to its Lean/Scribe/Markdown triple.

Each row preserves the original source path, SHA-256 and capacity span. Selected command excerpts within those spans are copied, with the Mathlib-derived OrderDense existence proof explicitly distinguished below; neither whole-file nor whole-span verbatim copying is asserted. Resolve each original path relative to the immutable upstream revision linked above. This is a source transplant, with no installed upstream package dependency or package-pin change.

The layer supplies logic symbols, semantic and forcing-relation interfaces, entailment and propositional calculi, predicate-language and term syntax, rewriting support and finite-data utilities for first-order developments. FormulaOne supplies the actual finite inductive first-order formula carrier with arbitrary relation arities, separate free variables and finite bound-variable indices, connectives, quantifiers, negation, structural recursion, complexity and conditional decidable equality. FormulaTwo adds finite free-variable support, bounds for natural-number variables and language maps with connective and quantifier preservation. This formula construction/support pair depends only on the frozen D5 Term/Quantifier APIs and pinned Mathlib; FormulaTwo also imports FormulaOne. These general APIs have independent mathematical use before the later term/formula rewriting and CSA graph-elimination layers.

References in the Scribe prose to a concrete pair interpretation describe intended downstream use. This layer does not install a pair-language or pair-interpretation interface, establish ZFC conservativity or a relative-consistency bridge, construct the full CSA, or prove ZFC model existence or absolute consistency. Internal HF/omega, finite archives, CSA arithmetic and rational/Cauchy-sequence constructions, and the complete CSA defining-graph and elimination obligations are outside this layer’s scope. It does not identify an internal nonstandard carrier with a Lean type or establish source-atom coverage.

The Empty/UniqueChoice support pair supplies equality of every map from an empty domain with its eliminator, and the chosen witness and specification of a proved unique existence on an arbitrary sort. It assumes no model of ZFC and does not establish full definitional conservativity. These are general API utilities (utility: none), with individually necessary rule-11 upstream-wrapper candidates and one compiler-generated exact-command companion. Generation supplies no new escape, proof-value, liveness or indispensability credit. The two Lean theorem names Classical.choose!_spec and Classical.choose!_eq_iff_right remain in prose with their typed formulas; choose_uniq is the legal direct selector. This pair has independent support meaning before the separate countable-filter and CSA interpretation layers. OrderDense supplies the countable dense-family filter API, including the empty family and membership of the prescribed initial element. The empty-family interface is the existence theorem specialized to the empty set; the separate Order.PFilter.IsGeneric.empty convenience instance is omitted. Its bundled density and genericity definitions follow Foundation. Its existence proof is a direct application of pinned Mathlib Order.idealOfCofinals, Order.mem_idealOfCofinals and Order.cofinal_meets_idealOfCofinals on the dual preorder, indexed by the family subtype using Set.Countable.toEncodable. Order.PFilter wraps the resulting ideal without changing its carrier. The local bridge is Mathlib-derived; it does not retain Foundation’s sequence, choice or descending-chain construction and asserts no identity of chosen filters. Mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d supplies Mathlib/Order/Ideal.lean, Mathlib/Order/PFilter.lean and Mathlib/Data/Set/Countable.lean under Apache-2.0, with attribution to their Mathlib contributors and source copyright notices. RewFour supplies seven lawful rewriting/substitution identities over arbitrary signatures. Both are retained prerequisite APIs; neither establishes the complete CSA defining graphs, definition elimination, conservativity or model existence.

Modifications are canonical headers, import reduction/relocation, source-command excerpts, capacity scope boundaries and restricting attribute target lists to needed relation projections. The retained Foundation command bodies remain upstream bytes, except for the OrderDense generic-filter existence proof described below.

The Semantics selection omits precisely the optional Semantics.Top (Set M) instance command at Foundation/Logic/Semantics.lean:251 (the upstream authored instance named LO.Semantics.instTopSet by Lean). The selected set_models_iff, set_meaningful_iff_nonempty and meaningful_iff_satisfiableSet commands retain their original proofs. In particular, the original simpa [NotModels, set_models_iff] remains in the nonemptiness proof. Compiler normalization helpers are generated from the retained proof source; their ownership does not establish a need for the omitted instance. The capacity span below locates the excerpts and includes this explicit exclusion.

The FormulaOne selection omits the optional upstream theorem LO.FirstOrder.Semiformula.neg_allClosure. The retained formula carrier and support commands preserve their original mathematical bodies and hypotheses; this exclusion belongs to the selected excerpts within FormulaOne’s capacity span below.

RewTwo supplies variable-domain lifts, binder shifts, shift, free, fix, their bound/free-variable laws, composition and substitution identities, q-lifts and finite fixitr iteration on actual first-order terms. It retains the supplied source commands and hypotheses except the optional LO.FirstOrder.Rew.q_emb. The supplied selection input is SHA-256 6639a4bd42da9b1cd99728d18f301d367f0a75c3afd0728373ce35c8ef55335c (6,268 bytes); the installed source, after that omission, is SHA-256 b68465527cc3e61f7f31322848f3421d60b533f8195861822bb7ed2b98a91652. These source digests are distinct from the complete upstream-file digest in the table. Historical 49-occurrence evidence associated with source SHA-256 d46397c4c5fb43b10a44c0596f5d7ab181e5e11a42ccd326611abe066825b2e8 is historical association evidence only; current identities require the current source-bound canonical report. Its 33 rule-11 upstream-wrapper proposals and 16 generated-source/API associations confer no new escape, liveness or indispensability credit. Theorems remain bind-only; definitions and other non-theorems have null proof shape. These general APIs do not prove the CSA set-coding, defining-graph elimination or ZFC conservativity bridge.

RewThree extends the term rewriting support with finite fixitr laws, term extensionality over free variables, language-map compatibility, free-variable propagation, conversion of closed terms to ClosedSemiterm, formula rewriting interfaces, notation and lawful syntactic rewriting identities. The supplied selection input is SHA-256 5b780be1e2900eca589caa86bf19ba20debb1253f048d784c08c96c114fd6fe9 (10,210 bytes), and the installed source has the same SHA-256. These declarations are retained as source-command excerpts from the capacity span below; their theorem observations are bind-only and the individually justified basis is rule-11-upstream-wrapper. No new escape, liveness or indispensability credit is claimed, and this support layer does not establish a pair interpretation, defining-graph elimination, ZFC conservativity or consistency/model-existence bridge.

Retirement: replace a transplanted API by a direct Mathlib reference when the repository’s pin provides a proved-equivalent syntax/theory/proof/definability API and its faithful bridge elaborates. Upstream PR acceptance alone is insufficient.

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