DI0006

divisor_complement_bounded

Divisor complementation stays in the exact inclusive interval 0..n, including its explicitly fixed omitted indices.

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 → Le(d,n) → DivisorComplement(n,d,q) → Le(q,n)

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

Definition DAG

DivisorComplement(n,d,q) · 1Le(a,b) · 2

Actual proof prerequisites

divisor_le_nonzero · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order statement 
forall n d q. ~(n=0) -> (exists pvs_le_gap_bounded_input. pvs_le_gap_bounded_input + (d) = (n)) -> ((((~((d)=0)) /\ ((n)=(d)*(q)))) \/ ((((d)=0 \/ ~(exists pvs_factor_bounded_graphnondivisor. (n) = (d) * pvs_factor_bounded_graphnondivisor)) /\ ((q)=(d))))) -> (exists pvs_le_gap_bounded_result. pvs_le_gap_bounded_result + (q) = (n))

Complete tactic proof in conservative notation

All 19 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

19 script commands · 9 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 hn
  5. L5
    intro hd
  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 factsL9–12

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

  1. L9
    specialize divisor_le_nonzero (q)
  2. L10
    specialize divisor_le_nonzero (n)
  3. L11
    apply divisor_le_nonzero
  4. L12
    exact hn
04Construct an explicit witnessL13–13

Supply the displayed value, then prove that it has the required property.

  1. L13
    exists d
05Calculate and transport equalitiesL14–14

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

  1. L14
    trans d*q
06Use earlier factsL15–16

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

  1. L15
    exact hq_left_right
  2. L16
    apply mul_comm
07Separate the logical casesL17–17

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

  1. L17
    cases hq_right
08Calculate and transport equalitiesL18–18

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

  1. L18
    rewrite hq_right_right
09Use earlier factsL19–19

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

  1. L19
    exact hd

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro n
  2. 0002intro d
  3. 0003intro q
  4. 0004intro hn
  5. 0005intro hd
  6. 0006intro hq
  7. 0007cases hq
  8. 0008cases hq_left
  9. 0009specialize divisor_le_nonzero (q)
  10. 0010specialize divisor_le_nonzero (n)
  11. 0011apply divisor_le_nonzero
  12. 0012exact hn
  13. 0013exists d
  14. 0014trans d*q
  15. 0015exact hq_left_right
  16. 0016apply mul_comm
  17. 0017cases hq_right
  18. 0018rewrite hq_right_right
  19. 0019exact hd