GI0002

signed_integer_floor_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every signed pair admits an actual floor quotient and strict remainder for every nonzero natural divisor; one ordinary natural division suffices.

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.

Exact expanded first-order arithmetic statement

forall xp xn m. ~(m = 0) -> exists qp qn r. (((xp) + (m) * (qn) = ((xn) + (m) * (qp)) + (r) /\ exists sif_gap_pair_total. sif_gap_pair_total + S (r) = (m)))

Constructive proof overview

Generated structural guide

Every signed pair admits an actual floor quotient and strict remainder for every nonzero natural divisor; one ordinary natural division suffices.

The unchanged tactic script uses 5 declared prerequisites and contains 30 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

nonzero_is_succ Stable theorem; checked-use authorized division_remainder_exists Stable theorem; checked-use authorized mul_succ_left Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

GI0004 signed_integer_canonical_floor_exists GI0039 gaussian_nearest_signed_quotient_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

30 script commands · 11 reading checkpoints · 2 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro xp
  2. L2
    intro xn
  3. L3
    intro m
  4. L4
    intro hm
02Establish hpredL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L5
    have hpred : exists h. m = S h
  2. L6
    specialize nonzero_is_succ m
  3. L7
    apply nonzero_is_succ
  4. L8
    exact hm
03Separate the logical casesL9–9

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

  1. L9
    cases hpred
04Establish hdivL10–14

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

  1. L10
    have hdiv : exists q r. xp + x * xn = m * q + r /\ (exists k. k + S r = m)
  2. L11
    specialize division_remainder_exists m
  3. L12
    specialize division_remainder_exists (xp + x * xn)
  4. L13
    apply division_remainder_exists
  5. L14
    exact hm
05Separate the logical casesL15–17

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

  1. L15
    cases hdiv
  2. L16
    cases hdiv_witness
  3. L17
    cases hdiv_witness_witness
06Construct an explicit witnessL18–20

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

  1. L18
    exists x1
  2. L19
    exists xn
  3. L20
    exists x2
07Separate the logical casesL21–21

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

  1. L21
    split
08Calculate and transport equalitiesL22–26

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

  1. L22
    trans (xp + x * xn) + xn
  2. L23
    rewrite hpred_witness
  3. L24
    simp [mul_succ_left, add_assoc]
  4. L25
    trans (m * x1 + x2) + xn
  5. L26
    congr
09Use earlier factsL27–27

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

  1. L27
    exact hdiv_witness_witness_left
10Calculate and transport equalitiesL28–29

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

  1. L28
    refl
  2. L29
    simp [add_assoc, add_comm]
11Use earlier factsL30–30

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

  1. L30
    exact hdiv_witness_witness_right

Library-wide reading audit

Original exact command ledger · 30 lines
  1. 0001intro xp
  2. 0002intro xn
  3. 0003intro m
  4. 0004intro hm
  5. 0005have hpred : exists h. m = S h
  6. 0006specialize nonzero_is_succ m
  7. 0007apply nonzero_is_succ
  8. 0008exact hm
  9. 0009cases hpred
  10. 0010have hdiv : exists q r. xp + x * xn = m * q + r /\ (exists k. k + S r = m)
  11. 0011specialize division_remainder_exists m
  12. 0012specialize division_remainder_exists (xp + x * xn)
  13. 0013apply division_remainder_exists
  14. 0014exact hm
  15. 0015cases hdiv
  16. 0016cases hdiv_witness
  17. 0017cases hdiv_witness_witness
  18. 0018exists x1
  19. 0019exists xn
  20. 0020exists x2
  21. 0021split
  22. 0022trans (xp + x * xn) + xn
  23. 0023rewrite hpred_witness
  24. 0024simp [mul_succ_left, add_assoc]
  25. 0025trans (m * x1 + x2) + xn
  26. 0026congr
  27. 0027exact hdiv_witness_witness_left
  28. 0028refl
  29. 0029simp [add_assoc, add_comm]
  30. 0030exact hdiv_witness_witness_right