TS000U · theorem body

affine_grid_point_remainder_exists

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ p. ∀ w. ∀ r. ∀ k. ¬p = 0 → ¬w = 0 → ∃ x. ∃ y. ∃ z. ∃ n. k = w · x + y ∧ (Lt(y,w) ∧ DivRem(r · x + y,p,z,n))

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0007 DivRem

In local proof propositions

PD0007 DivRem
Exact expanded first-order 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))))))

Proof neighborhood

Direct theorem prerequisites

division_remainder_exists · Stable closed

Direct theorem dependents

TS000W beta_affine_residue_grid_exists

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

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 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 : ∃ i. ∃ j. DivRem(k,w,i,j)Definitions: DivRem(k,w,i,j)Original native command in the exact edition
  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 : ∃ q. ∃ t. DivRem(r · x + x1,p,q,t)Definitions: DivRem(r · x + x1,p,q,t)Original native command in the exact edition
  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 defined 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 : ∃ i. ∃ j. DivRem(k,w,i,j)
    Exact native replay linehave 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 : ∃ q. ∃ t. DivRem(r · x + x1,p,q,t)
    Exact native replay linehave 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