DI0005

divisor_complement_symmetric

For positive n the actual complementary quotient is reversible; zero and nondivisors remain fixed.

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

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

Definition DAG

DivisorComplement(n,d,q) · 2

Actual proof prerequisites

factor_nonzero_right · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order statement 
forall n d q. ~(n=0) -> ((((~((d)=0)) /\ ((n)=(d)*(q)))) \/ ((((d)=0 \/ ~(exists pvs_factor_symmetric_sourcenondivisor. (n) = (d) * pvs_factor_symmetric_sourcenondivisor)) /\ ((q)=(d))))) -> ((((~((q)=0)) /\ ((n)=(q)*(d)))) \/ ((((q)=0 \/ ~(exists pvs_factor_symmetric_targetnondivisor. (n) = (q) * pvs_factor_symmetric_targetnondivisor)) /\ ((d)=(q)))))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 11 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 assumptionsL1–5

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 hn
  5. L5
    intro hq
02Separate the logical casesL6–9

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

  1. L6
    cases hq
  2. L7
    cases hq_left
  3. L8
    left
  4. L9
    split
03Fix variables and assumptionsL10–10

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

  1. L10
    intro hqzero
04Use earlier factsL11–17

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

  1. L11
    specialize factor_nonzero_right (n)
  2. L12
    specialize factor_nonzero_right (d)
  3. L13
    specialize factor_nonzero_right (q)
  4. L14
    apply factor_nonzero_right
  5. L15
    exact hn
  6. L16
    exact hq_left_right
  7. L17
    exact hqzero
05Calculate and transport equalitiesL18–18

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L18
    trans d*q
06Use earlier factsL19–20

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

  1. L19
    exact hq_left_right
  2. L20
    apply mul_comm
07Separate the logical casesL21–21

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

  1. L21
    cases hq_right
08Calculate and transport equalitiesL22–26

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L22
    rewrite hq_right_right
  2. L23
    rewrite hq_right_right
  3. L24
    rewrite hq_right_right
  4. L25
    rewrite hq_right_right
  5. L26
    rewrite hq_right_right
09Separate the logical casesL27–28

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

  1. L27
    right
  2. L28
    split
10Use earlier factsL29–29

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

  1. L29
    exact hq_right_left
11Calculate and transport equalitiesL30–30

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L30
    refl

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro n
  2. 0002intro d
  3. 0003intro q
  4. 0004intro hn
  5. 0005intro hq
  6. 0006cases hq
  7. 0007cases hq_left
  8. 0008left
  9. 0009split
  10. 0010intro hqzero
  11. 0011specialize factor_nonzero_right (n)
  12. 0012specialize factor_nonzero_right (d)
  13. 0013specialize factor_nonzero_right (q)
  14. 0014apply factor_nonzero_right
  15. 0015exact hn
  16. 0016exact hq_left_right
  17. 0017exact hqzero
  18. 0018trans d*q
  19. 0019exact hq_left_right
  20. 0020apply mul_comm
  21. 0021cases hq_right
  22. 0022rewrite hq_right_right
  23. 0023rewrite hq_right_right
  24. 0024rewrite hq_right_right
  25. 0025rewrite hq_right_right
  26. 0026rewrite hq_right_right
  27. 0027right
  28. 0028split
  29. 0029exact hq_right_left
  30. 0030refl