{"slug":"formal-exponent-multiplication-parity","identifier":"formal-exponent-multiplication-parity","title":"Exponent Multiplication Parity","author":"Maclain Hunter","sourceSha256":"84fe5492d3f527cd1c959228d59ea71762363696064d4226925d2756b3dc8b67","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":"2026-10-03T10:38:14.008Z","status":"partial","axiomFree":false,"axioms":["propext","Classical.choice","Quot.sound"],"theoremNames":["HunterAuthored.parity_power_multiplication"],"units":[{"id":"parity_power_multiplication","sourceStatement":"Formal statement: (-1)^a * (-1)^b = (-1)^(a+b)","sourceOffset":0,"scope":"Unbounded integer parity-defined powers multiply by exponent addition. The dependency report explicitly retains the arithmetic tactic's kernel axioms; this theorem is not automatically labeled axiom-free.","status":"candidate","variables":[],"premises":[],"conclusion":"See the exact typed Lean theorem","leanStatement":"def parityPower (n : Int) : Int := if n % 2 = 0 then 1 else -1\ntheorem parity_power_multiplication (a b : Int) :\n    parityPower a * parityPower b = parityPower (a + b) := by\n  unfold parityPower\n  have ra : a % 2 = 0 ∨ a % 2 = 1 := by omega\n  have rb : b % 2 = 0 ∨ b % 2 = 1 := by omega\n  rcases ra with ha | ha <;> rcases rb with hb | hb\n  · have hc : (a + b) % 2 = 0 := by omega\n    rw [if_pos ha, if_pos hb, if_pos hc]; rfl\n  · have hc : (a + b) % 2 ≠ 0 := by omega\n    have hn : b % 2 ≠ 0 := by omega\n    rw [if_pos ha, if_neg hn, if_neg hc]; rfl\n  · have hc : (a + b) % 2 ≠ 0 := by omega\n    have hn : a % 2 ≠ 0 := by omega\n    rw [if_neg hn, if_pos hb, if_neg hc]; rfl\n  · have hc : (a + b) % 2 = 0 := by omega\n    have hn : a % 2 ≠ 0 := by omega\n    have hm : b % 2 ≠ 0 := by omega\n    rw [if_neg hn, if_neg hm, if_pos hc]; rfl\n#print axioms parity_power_multiplication\n","kernelStatus":"verified","axioms":["propext","Classical.choice","Quot.sound"],"axiomFree":false,"reportedTheorem":"HunterAuthored.parity_power_multiplication"}],"files":{"source.json":{"sha256":"fbb51e269cafade39432ff2af6ca0998c3bf52a4cf79d4d1b2a40bfab52c07ef","bytes":672},"Proof.lean":{"sha256":"72a16f1edcb5b6c8f1c9e3b1a04b8371f53b10fe102802ea326cb5ffc45e439c","bytes":1180},"lean-check.txt":{"sha256":"ac12f6193418d720ecc6d2a16d4b9c997663c9ff44988857f0ae9df531b235a0","bytes":160}},"boundary":"Lean establishes the listed formal statements and dependencies. 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