{"slug":"formal-and-commutativity","identifier":"formal-and-commutativity","title":"AND Commutativity","author":"Maclain Hunter","sourceSha256":"8c5b22fe03debca9b8a0f493eed18819959fa416ddc0b7987424033a163fbad1","sourceKind":"formal-theorem","recordedAt":null,"recordedAtMeaning":"Database registration time, not a claim of original handwritten authorship.","originalAuthoredAt":null,"formalizedAt":"2026-10-03T10:37:05.082Z","checkedAt":"2026-10-03T10:37:23.516Z","status":"partial","axiomFree":false,"axioms":["propext"],"theoremNames":["HunterAuthored.magnitude_and_commutative"],"units":[{"id":"magnitude_and_commutative","sourceStatement":"Formal statement: ∀a,b ∈ MCBExponent: AND(a,b) = AND(b,a)","sourceOffset":0,"scope":"Commutativity of the displayed integer sign-times-recursive-minimum-magnitude realization. The minimum and zero behavior are defined explicitly; actual library dependencies are retained in the report.","status":"candidate","variables":[],"premises":[],"conclusion":"See the exact typed Lean theorem","leanStatement":"def minimumNat : Nat → Nat → Nat\n  | 0, _ => 0\n  | _, 0 => 0\n  | a + 1, b + 1 => minimumNat a b + 1\ntheorem minimum_commutative (a b : Nat) : minimumNat a b = minimumNat b a := by\n  induction a generalizing b with\n  | zero => cases b <;> rfl\n  | succ a ih =>\n    cases b with\n    | zero => rfl\n    | succ b => exact congrArg Nat.succ (ih b)\ndef magnitudeAnd (a b : Int) : Int :=\n  (if a * b < 0 then -1 else 1) * (minimumNat a.natAbs b.natAbs : Int)\ntheorem magnitude_and_commutative (a b : Int) : magnitudeAnd a b = magnitudeAnd b a := by\n  unfold magnitudeAnd\n  rw [Int.mul_comm a b, minimum_commutative a.natAbs b.natAbs]\n#print axioms magnitude_and_commutative\n","kernelStatus":"verified","axioms":["propext"],"axiomFree":false,"reportedTheorem":"HunterAuthored.magnitude_and_commutative"}],"files":{"source.json":{"sha256":"e8b3f6699b4cdcbe181fe69646352885a2297dfdb1b5df99040da21cf4731ad2","bytes":754},"Proof.lean":{"sha256":"226f7ac491e2d11ae7c7365222d6ab158a95364edfcd056a7d322697eb9a90e5","bytes":938},"lean-check.txt":{"sha256":"b48de3ce8f3a4c039376421047b5099f663ef71fe02524eeb32ab9a4ab3245d1","bytes":128}},"boundary":"Lean establishes the listed formal statements and dependencies. The exact authored source, Ground, known premise meanings, application validation and empirical evidence remain separately typed. No unencoded section is certified by a neighboring theorem.","currentSourceSha256":"8c5b22fe03debca9b8a0f493eed18819959fa416ddc0b7987424033a163fbad1","timestamp":"2026-10-03T10:42:55.000Z","timestampAuthority":"https://www.freetsa.org","manifestSha256":"6e4e8d5c750a86f20e8f987a742b4eb2f588563a5f23a438ab84950f16eec8c4","url":"/proof-verification/formal-and-commutativity","jsonUrl":"/proof-verification/formal-and-commutativity.json","leanUrl":"/proof-verification/formal-and-commutativity.lean","graph":{"@context":"https://schema.org","@type":"SoftwareSourceCode","@id":"https://4p-engine.com/proof-verification/formal-and-commutativity#lean","name":"Lean 4 witness: AND Commutativity","url":"https://4p-engine.com/proof-verification/formal-and-commutativity","dateCreated":"2026-10-03T10:37:05.082Z","dateModified":"2026-10-03T10:37:23.516Z","creator":{"@type":"Organization","name":"Replit Agent"},"programmingLanguage":{"@type":"ComputerLanguage","name":"Lean","version":"Lean (version 4.7.0, commit v4.7.0, Release)"},"author":{"@type":"Organization","name":"Replit Agent"},"description":"Lean establishes the listed formal statements and dependencies. The exact authored source, Ground, known premise meanings, application validation and empirical evidence remain separately typed. No unencoded section is certified by a neighboring theorem.","isBasedOn":{"@type":"CreativeWork","name":"AND Commutativity","author":{"@type":"Person","name":"Maclain Hunter"},"url":"https://4p-engine.com/proof-knowledge/formal-and-commutativity","identifier":"formal-and-commutativity"},"encoding":{"@type":"MediaObject","encodingFormat":"text/plain","contentUrl":"https://4p-engine.com/proof-verification/formal-and-commutativity.lean"},"subjectOf":{"@type":"DigitalDocument","name":"Independently verified RFC 3161 existence timestamp","dateCreated":"2026-10-03T10:42:55.000Z","url":"https://4p-engine.com/proof-verification/timestamp.json"},"additionalProperty":[{"@type":"PropertyValue","name":"verificationStatus","value":"partial"},{"@type":"PropertyValue","name":"axiomFree","value":"false"},{"@type":"PropertyValue","name":"axiomDependencies","value":"[\"propext\"]"},{"@type":"PropertyValue","name":"theorems","value":"[\"HunterAuthored.magnitude_and_commutative\"]"},{"@type":"PropertyValue","name":"sourceSha256","value":"8c5b22fe03debca9b8a0f493eed18819959fa416ddc0b7987424033a163fbad1"},{"@type":"PropertyValue","name":"releaseManifestSha256","value":"6e4e8d5c750a86f20e8f987a742b4eb2f588563a5f23a438ab84950f16eec8c4"},{"@type":"PropertyValue","name":"sourceRecordedAt","value":"null"},{"@type":"PropertyValue","name":"sourceDateMeaning","value":"Database registration time, not a claim of original handwritten authorship."},{"@type":"PropertyValue","name":"originalAuthoredAt","value":"null"},{"@type":"PropertyValue","name":"timestampAuthority","value":"https://www.freetsa.org"},{"@type":"PropertyValue","name":"timestampScope","value":"Independent cryptographic existence timestamp for the manifest and its content-addressed artifacts. Not legal notarization, an authorship adjudication, or a retroactive timestamp for handwritten work."},{"@type":"PropertyValue","name":"governingGround","value":"Logic / Logos / God"},{"@type":"PropertyValue","name":"formalizationAttribution","value":"Machine formalization by Replit Agent of Maclain Hunter's unchanged authored source"}]}}