{"slug":"archived-q1055-1bac66106909","identifier":"archived-q1055-1bac66106909","title":"AND Commutativity","author":"Maclain Hunter","sourceSha256":"1bac6610690928014d0382c506eed874cfb6aef628097814e581ffe09d8bc95a","sourceKind":"archived-authored-source","recordedAt":null,"recordedAtMeaning":"Database registration time, not a claim of original handwritten authorship.","originalAuthoredAt":null,"formalizedAt":"2026-10-03T10:37:05.082Z","checkedAt":"2026-10-03T10:37:24.380Z","status":"partial","axiomFree":false,"axioms":["propext"],"theoremNames":["HunterAuthored.magnitude_and_commutative"],"units":[{"id":"magnitude_and_commutative","sourceStatement":"{\n  \"name\": \"AND Commutativity\",\n  \"formal\": \"∀a,b ∈ MCBExponent: AND(a,b) = AND(b,a)\",\n  \"proof\": \"Follows from commutativity of multiplication and symmetry of min\",\n  \"implementation\": \"mcbExponentAND(a: MCBExponent, b: MCBExponent): MCBExponent\"\n}","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":"1cde377c2c1d54feb4a672b34511bb893682f640d3e8ff2bee749c1e11485921","bytes":803},"Proof.lean":{"sha256":"b94376f96a60756e26914e744f3b7651f0bf9d9e675e6489d0c1e93934bae56f","bytes":938},"lean-check.txt":{"sha256":"b48de3ce8f3a4c039376421047b5099f663ef71fe02524eeb32ab9a4ab3245d1","bytes":128}},"boundary":"Lean establishes the listed formal statements and dependencies. 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