TS000U

affine_grid_point_remainder_exists

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

Every flat grid index has canonical row, column, affine quotient, and strictly bounded residue witnesses.

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 p w r k. ~(p = 0) -> ~(w = 0) -> (exists ftrg_row_point ftrg_column_point ftrg_quotient_point ftrg_remainder_point. ((k) = (w) * ftrg_row_point + ftrg_column_point /\ ((exists ftrg_gap_point_column. ftrg_gap_point_column + S (ftrg_column_point) = (w)) /\ ((r * ftrg_row_point + ftrg_column_point = (p) * ftrg_quotient_point + ftrg_remainder_point) /\ (exists ftrg_gap_point_residue. ftrg_gap_point_residue + S (ftrg_remainder_point) = (p))))))

Constructive proof overview

Generated structural guide

Every flat grid index has canonical row, column, affine quotient, and strictly bounded residue witnesses.

The unchanged tactic script uses 1 declared prerequisite and contains 33 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

division_remainder_exists Stable theorem; checked-use authorized

Direct dependents

TS000W beta_affine_residue_grid_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

33 script commands · 12 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 assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro w
  3. L3
    intro r
  4. L4
    intro k
  5. L5
    intro hp
  6. L6
    intro hw
02Establish hcoordinatesL7–11

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

  1. L7
    have hcoordinates : exists i j. k = w * i + j /\ exists gap. gap + S j = w
  2. L8
    specialize division_remainder_exists w
  3. L9
    specialize division_remainder_exists k
  4. L10
    apply division_remainder_exists
  5. L11
    exact hw
03Separate the logical casesL12–14

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

  1. L12
    cases hcoordinates
  2. L13
    cases hcoordinates_witness
  3. L14
    cases hcoordinates_witness_witness
04Establish hresidueL15–19

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

  1. L15
    have hresidue : exists q t. r * x + x1 = p * q + t /\ exists gap. gap + S t = p
  2. L16
    specialize division_remainder_exists p
  3. L17
    specialize division_remainder_exists (r * x + x1)
  4. L18
    apply division_remainder_exists
  5. L19
    exact hp
05Separate the logical casesL20–22

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

  1. L20
    cases hresidue
  2. L21
    cases hresidue_witness
  3. L22
    cases hresidue_witness_witness
06Construct an explicit witnessL23–26

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

  1. L23
    exists x
  2. L24
    exists x1
  3. L25
    exists x2
  4. L26
    exists x3
07Separate the logical casesL27–27

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

  1. L27
    split
08Use earlier factsL28–28

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

  1. L28
    exact hcoordinates_witness_witness_left
09Separate the logical casesL29–29

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

  1. L29
    split
10Use earlier factsL30–30

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

  1. L30
    exact hcoordinates_witness_witness_right
11Separate the logical casesL31–31

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

  1. L31
    split
12Use earlier factsL32–33

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

  1. L32
    exact hresidue_witness_witness_left
  2. L33
    exact hresidue_witness_witness_right

Library-wide reading audit

Original exact command ledger · 33 lines
  1. 0001intro p
  2. 0002intro w
  3. 0003intro r
  4. 0004intro k
  5. 0005intro hp
  6. 0006intro hw
  7. 0007have hcoordinates : exists i j. k = w * i + j /\ exists gap. gap + S j = w
  8. 0008specialize division_remainder_exists w
  9. 0009specialize division_remainder_exists k
  10. 0010apply division_remainder_exists
  11. 0011exact hw
  12. 0012cases hcoordinates
  13. 0013cases hcoordinates_witness
  14. 0014cases hcoordinates_witness_witness
  15. 0015have hresidue : exists q t. r * x + x1 = p * q + t /\ exists gap. gap + S t = p
  16. 0016specialize division_remainder_exists p
  17. 0017specialize division_remainder_exists (r * x + x1)
  18. 0018apply division_remainder_exists
  19. 0019exact hp
  20. 0020cases hresidue
  21. 0021cases hresidue_witness
  22. 0022cases hresidue_witness_witness
  23. 0023exists x
  24. 0024exists x1
  25. 0025exists x2
  26. 0026exists x3
  27. 0027split
  28. 0028exact hcoordinates_witness_witness_left
  29. 0029split
  30. 0030exact hcoordinates_witness_witness_right
  31. 0031split
  32. 0032exact hresidue_witness_witness_left
  33. 0033exact hresidue_witness_witness_right