DI0004

divisor_complement_positive_equation

At every genuine positive divisor the complement output satisfies n=d*q; the identity convention cannot supply that branch.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

The complementary quotient is witnessed by n=d*q at positive divisors. The actual permutation covers indices zero through n, fixing zero and nondivisors. This is the involution foundation for the separate cancellation and full G007 inversion proofs, not an assumed divisor bijection.

Exact theorem in conservative defined notation

∀ n. ∀ d. ∀ q. ¬d = 0 → Dvd(d,n) → DivisorComplement(n,d,q) → n = d · q

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

DivisorComplement(n,d,q) · 1Dvd(d,n) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall n d q. ~(d=0) -> (exists pvs_factor_equation_divisor. (n) = (d) * pvs_factor_equation_divisor) -> ((((~((d)=0)) /\ ((n)=(d)*(q)))) \/ ((((d)=0 \/ ~(exists pvs_factor_equation_graphnondivisor. (n) = (d) * pvs_factor_equation_graphnondivisor)) /\ ((q)=(d))))) -> n=d*q

Complete tactic proof in conservative notation

All 16 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

16 script commands · 5 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–6

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro n
  2. L2
    intro d
  3. L3
    intro q
  4. L4
    intro hd
  5. L5
    intro hdiv
  6. L6
    intro hq
02Separate the logical casesL7–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    cases hq
  2. L8
    cases hq_left
03Use earlier factsL9–9

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L9
    exact hq_left_right
04Separate the logical casesL10–12

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L10
    cases hq_right
  2. L11
    exfalso
  3. L12
    cases hq_right_left
05Use earlier factsL13–16

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L13
    apply hd
  2. L14
    exact hq_right_left_left
  3. L15
    apply hq_right_left_right
  4. L16
    exact hdiv

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro n
  2. 0002intro d
  3. 0003intro q
  4. 0004intro hd
  5. 0005intro hdiv
  6. 0006intro hq
  7. 0007cases hq
  8. 0008cases hq_left
  9. 0009exact hq_left_right
  10. 0010cases hq_right
  11. 0011exfalso
  12. 0012cases hq_right_left
  13. 0013apply hd
  14. 0014exact hq_right_left_left
  15. 0015apply hq_right_left_right
  16. 0016exact hdiv