GI0002

signed_integer_floor_exists

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

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 natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.

Exact theorem in conservative defined notation

∀ xp. ∀ xn. ∀ m. ¬m = 0 → ∃ x. ∃ y. ∃ z. SignedFloor(xp,xn,m,x,y,z)

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

Definition DAG

SignedFloor(p,n,m,q,t,r) · 1Lt(a,b) · 1

Actual proof prerequisites

nonzero_is_succ · checked external prerequisitedivision_remainder_exists · checked external prerequisitemul_succ_left · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order 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)))

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 · 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.

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 assumptionsL1–4

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 : ∃ q. ∃ r. xp + x · xn = m · q + r ∧ Lt(r,m)Definitions: Lt(r,m)Original native command in the exact edition
  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 defined 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 : ∃ q. ∃ r. xp + x · xn = m · q + r ∧ Lt(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