AF0001

foundation_division_exists_unique

G001: construct the quotient and strict remainder for every nonzero divisor and prove the pair is literally unique.

Alpha v34 checked-use · first admitted v28 · 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. Exact original first-admission records.

The original division, gcd, cancellation, and factor-existence foundations are exposed through checked wrappers. Unordered uniqueness adds an actual bounded, injective, surjective index map matching repeated prime occurrences. The empty factor list represents one, not zero.

Exact theorem in conservative defined notation

∀ n. ∀ d. ¬d = 0 → ∃ x. ∃ y. DivRem(n,d,x,y) ∧ (∀ z. ∀ m. DivRem(n,d,z,m) → z = x ∧ m = y)

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

Definition DAG

DivRem(n,d,q,r) · 3

Actual proof prerequisites

division_remainder_exists · checked external prerequisitedivision_remainder_unique · checked external prerequisite
Original expanded first-order statement 
forall n d. ~(d = 0) -> exists q r. (((n = d * q + r /\ (exists fsat_gap_chosen. fsat_gap_chosen + S (r) = (d)))) /\ forall u v. ((n = d * u + v /\ (exists fsat_gap_other. fsat_gap_other + S (v) = (d)))) -> u = q /\ v = r)

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 · 10 reading checkpoints · 1 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 assumptionsL1–3

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

  1. L1
    intro n
  2. L2
    intro d
  3. L3
    intro hd
02Establish hdivisionL4–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder exists.

  1. L4
    have hdivision : ∃ q. ∃ r. DivRem(n,d,q,r)Definitions: DivRem(n,d,q,r)Original native command in the exact edition
  2. L5
    specialize division_remainder_exists (d)
  3. L6
    specialize division_remainder_exists (n)
  4. L7
    apply division_remainder_exists
  5. L8
    exact hd
03Separate the logical casesL9–10

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

  1. L9
    cases hdivision
  2. L10
    cases hdivision_witness
04Construct an explicit witnessL11–12

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

  1. L11
    exists x
  2. L12
    exists x1
05Separate the logical casesL13–13

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

  1. L13
    split
06Use earlier factsL14–14

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

  1. L14
    exact hdivision_witness_witness
07Fix variables and assumptionsL15–17

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

  1. L15
    intro u
  2. L16
    intro v
  3. L17
    intro hother
08Separate the logical casesL18–19

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

  1. L18
    cases hdivision_witness_witness
  2. L19
    cases hother
09Use earlier factsL20–29

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

  1. L20
    specialize division_remainder_unique (d)
  2. L21
    specialize division_remainder_unique (n)
  3. L22
    specialize division_remainder_unique (u)
  4. L23
    specialize division_remainder_unique (v)
  5. L24
    specialize division_remainder_unique (x)
  6. L25
    specialize division_remainder_unique (x1)
  7. L26
    apply division_remainder_unique
  8. L27
    exact hother_left
  9. L28
    exact hother_right
  10. L29
    exact hdivision_witness_witness_left
10Use earlier factsL30–30

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

  1. L30
    exact hdivision_witness_witness_right

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro n
  2. 0002intro d
  3. 0003intro hd
  4. 0004have hdivision : ∃ q. ∃ r. DivRem(n,d,q,r)
  5. 0005specialize division_remainder_exists (d)
  6. 0006specialize division_remainder_exists (n)
  7. 0007apply division_remainder_exists
  8. 0008exact hd
  9. 0009cases hdivision
  10. 0010cases hdivision_witness
  11. 0011exists x
  12. 0012exists x1
  13. 0013split
  14. 0014exact hdivision_witness_witness
  15. 0015intro u
  16. 0016intro v
  17. 0017intro hother
  18. 0018cases hdivision_witness_witness
  19. 0019cases hother
  20. 0020specialize division_remainder_unique (d)
  21. 0021specialize division_remainder_unique (n)
  22. 0022specialize division_remainder_unique (u)
  23. 0023specialize division_remainder_unique (v)
  24. 0024specialize division_remainder_unique (x)
  25. 0025specialize division_remainder_unique (x1)
  26. 0026apply division_remainder_unique
  27. 0027exact hother_left
  28. 0028exact hother_right
  29. 0029exact hdivision_witness_witness_left
  30. 0030exact hdivision_witness_witness_right